selariu supermathematics functions, editor florentin smarandache

Post on 30-May-2018

216 views

Category:

Documents

0 download

Embed Size (px)

TRANSCRIPT

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    1/133

    Techno-Art of Selariu SuperMathematics Functions

    A e r o n a u t i c s C a p s u l e

    2007

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    2/133

    1

    Techno-Art of Selariu SuperMathematics Functions

    Editor: Florentin Smarandache

    ARP2007

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    3/133

    2

    Front cover art image, called Aeronautics Capsule, by Mircea Eugen elariu.

    This book can be ordered in a paper bound reprint from:

    Books on DemandProQuest Information & Learning

    (University of Microfilm International)

    300 N. Zeeb RoadP.O. Box 1346, Ann Arbor

    MI 48106-1346, USA

    Tel.: 1-800-521-0600 (Customer Service)http://wwwlib.umi.com/bod/basic

    Copyright 2007 by American Research Press, Editor and the Authors for their SuperMathematics

    Functions and Images.

    Many books can be downloaded from the following

    Digital Library of Arts & Letters:

    http://www.gallup.unm.edu/~smarandache/eBooksLiterature.htm

    Peer Reviewers:1. Prof. Mihly Bencze, University of Braov, Romania.

    2. Prof. Valentin Boju, Ph. D.

    Officer of the Order Cultural Merit, Category Scientific ResearchMontrealTech - Institut de Technologie de Montral

    Director, MontrealTech Press

    P. O. Box 78574, Station Wilderton

    Montral, Qubec, H3S 2W9, Canada

    (ISBN-10): 1-59973-037-5

    (ISBN-13): 978-1-59973-037-0

    (EAN): 9781599730370

    Printed in the United States of America

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    4/133

    3

    C-o-n-t-e-n-t-s

    A e r o n a u t i c s c a p s u l e first cover

    FOREWORD (FOR SUPERMATHEMATICS FUNCTIONS) 6-16

    Selariu SuperMathematics Functions

    & Other SuperMathematics Functions 17

    D O U B L E C L E P S Y D R A 18S U P E R M A T H E M A T I C S F L O W E R S

    19The Ballet of the Functions 20

    J a c u z z i 21D a m a g e d p a r t o f T I T A N I C 22K A Z A T C I O K (Russian Popular Dance) 23F l y i n g B i r d 1 24F l y i n g B i r d 2 25M U L T I C O L O R E D S U N 26R e d S u n 27T h e d o u b l e N o z z l e f o r N A S A 28T h e s u p e r m a t h e m a t i c s C o m e t 29The Lake of Swans 30

    + The Dance of Swords 30+ The Nut Cracker 30+ The Decease of Swan 30

    T h e F l o w e r i n g 31The supermathematics ring surface 32The ex-centric sphere 33T h e s u p e r m a t h e m a t i c s Screw Surface 34T h e T r o j a n H o r s e 35T h e A m p h o r a s 36B U D D H A 37J E T P L A N E 38T R O U B L E D L A N D

    o r D O U B L E A N A L Y T I C A L E X C E N T R I CF U N C T I O N 39

    S E L F - P I E R C E B O D Y 40H I L L S a n d V A L L E Y S 41B e r n o u l l i s L e m n i s c a t e, C a s s i n i s O v a l s a n d o t h e r s 42Continuous transforming of a circle into a haystack 43C y c l i c a l S y m m e t r y 44S m a r a n d a c h e S t e p p e d F u n c t i o n s 45S C R I B B L I N G S W I T H . . . H E A D A N D T A L E 46Q U A D R I P O D 47E X C E N T R I C S Y M M E T R Y 48

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    5/133

    4

    D R A C U L A S C A S T L E 49T U N I N G F O R K 50

    d e x O I D S + r e x O I D S 51E x c e n t r i c g e o m e t r y , p r i s m a t i c s o l i d s 52V a s e 53H E X A G O N A L T O R U S 54O p e n s q u a r e t o r u s 55D o u b l e s q u a r e t o r u s 56S i n u o u s C o r r u g a t e W a s h e r s

    o r N a n o - P e r i s t a l t i c E n g i n e 57I G L O O + T h e m a g i c c a r p e t 58Multiple Ex Centric Circular SuperMathematics Functions 59E X C E N T R I C T O R U S R I N G 60H Y P E R S O N I C J E T A I R P L A N E 61P L U M P V A S E 62A R R O W S 63H Y P E R B O L I C Q U A D R A T I C C Y L I N D E R 1 64H Y P E R B O L I C Q U A D R A T I C C Y L I N D E R 2 65E X - C E N T R I C F U L L S P R I N G 66E X - C E N T R I C E M P T Y S P R I N G 67E X C E N T R I C P E N T A G O N H E L I X 68C Y L I N D E R S w i t h C O L L A R S 69U n i c u r s a l S u p e r m a t h e m a t i c s F u n c t i o n s 1

    + O l d w o m a n f r o m C a r p a t i M o u n t a i n ( R o m a ni a ) 70U n i c u r s a l S u p e r m a t h e m a t i c s F u n c t i o n s 2 71U n i c u r s a l Su p e r m a t h e m a t i c s F u n c t i o n s 3

    + W a l k i n g P i n g u i n s 72T o D o u b l e a n d S i m p l e C a n o e 73R o m a n i a n f o l k d a n c e 74P i l l o w 75T e r r a w i t h M a r k e d M e r i d i a n s 76S u p e r m a t h e m a t i c s C o l u m n s 77A e r o d y n a m i c S o l i d 78H a l v i n g C u r v e 79The crook lines (s 0) - a generalization of straight lines ( s = 0 ) 80-82A R A B E S Q U E S 1 83S T A R S 84

    H y s t e r e t i c C u r v e s 1 85H y s t e r e t i c C u r v e s 2 86H y s t e r e t i c C u r v e s 3 87E x C e n t r i c C i r c u l a r C u r v e s w i t h E x C e n t r i c V a r i a b l e 88F i l i g r e e 1 89F I L I G R E E 2 90U s h a p e d C u r v e s i n 2 D a n d 3 D 91E x p l o s i o n s 92S p a t i a l F i g u r e 93P l a n e t s a n d s t a r s 94E x C e n t r i c C i r c l e ( n = 1 ) a n d A s t e r o i d ( n = 2, 4 , 6 ) 95E x C e n t r i c A s t e r o i d ( n = 3, 5 ,7 ,9) 96E x C e n t r i c L e m n i s c a t e s 97B u t t e r f l y w i t h S y m m e t r i c a l C e n t e r 1 98B u t t e r f l y w i t h S y m m e t r i c a l C e n t e r 2 99B u t t e r f l y w i t h S y m m e t r i c a l C e n t e r 3 100B u t t e r f l y R a p i d l y F l a p p i n g t h e W i n g s 101F l o w e r w i t h F o u r P e t a l e s 102

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    6/133

    5

    E c C e n t r i c P y r a m i d 103A e r o d y n a m i c P r o f i l e

    w i t h S u p e r m a t h e m a t i c s F u n c t i o n s 1 104A e r o d y n a m i c P r o f i l e

    w i t h S u p e r m a t h e m a t i c s F u n c t i o n s 2 105S A L T C e l l a r 106S U P E R M A T H E M A T I C S T O W E R 107THE CONTINUOUS TRANSFORMATION OF A RIGHT TRIANGLE INTO

    ITS HYPOTENUSE

    + THE CONTINUOUS TRANSFORMATION OF QUADRATE (Rx = Ry)

    OR RECTANGLE (Rx Ry) INTO ITS DIAGONAL 108Mo d i f i e d c e x a n d s e x 109S i n u o u s S u r f a c e w i t h A n a l i t i c a l S u p e r m a t h e m a t i c s

    F u n c t i o n s 110S u p e r m a t h e m a t i c s S p i r a l

    + S u p e r m a t h e m a t i c s P a r a b l e s 111A R A B E S Q U E S 2 112S u p e r m a t h e m a t i c s f u n c t i o n s cex xy and sex xy 113S u p e r m a t h e m a t i c s f u n c t i o n s rex xy and dex xy 114E x C e n t r i c F o l k l o r e C a r p e t 1 115E x C e n t r i c F o l k l o r e C a r p e t 2 116E x C e n t r i c F o l k l o r e C a r p e t 3 117W A T E R F A L L I N G 118

    S I N G L E and D O U B L E K C Y L I N D E R 119S U P E R M A T H E M A T I C A L K N O T S H A P E D B R E A D

    a n d O N E C R A C K N E L ( P R E T Z E L) 120S i x C o n o p y r a m i d s 121F O U R C O N O P Y R A M I D S V I E W E D F R O M A B O V E 122D O U B L E C O N O P Y R A M I D

    or the transformation of circle into a square with circular ex-centric supermathematics function dex orquadrilobic functions cosq and sinq v 123

    E X - C E N T R I C S a n d V A L E R I U A L A C I C U A D R O L O B S 124P E R F E C T C U B E 125E x c e n t r i c c i r c u l a r c u r v e s 1 126E x c e n t r i c c i r c u l a r c u r v e s 2

    o r E x c e n t r i c L i s s a j o u s c u r v e s 127

    References in SuperMathematics 128Postface back cover

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    7/133

    6

    FOREWORD

    (FOR SUPER-MATHEMATICS FUNCTIONS)

    In this album we include the so called Super-Mathematics functions (SMF), which

    constitute the base for, most often, generating, technical, neo-geometrical objects, therefore lessartistic.These functions are the results of 38 years of research, which began at University of Stuttgart in

    1969. Since then, 42 related works have been published, written by over 19 authors, as shown in theReferences.

    The name was given by the regretted mathematician Professor Emeritus Doctor Engineer

    Gheorghe Silas who, at the presentation of the very first work in this domain, during the First National Conference of Vibrations in Machine Constructions, Timioara, Romania, 1978, namedCIRCULAR EX-CENTRIC FUNCTIONS, declared: Young man, you just discovered not only

    some functions, but a new mathematics, a supermathematics! I was glad, at my age of 40, like ateenager. And I proudly found that he might be right!

    The prefix super is justified today, to point out the birth of the new complements inmathematics, joined together under the name of Ex-centric Mathematics (EM), with much more

    important and infinitely more numerous entities than the existing ones in the actual mathematics,which weare obliged to call it Centric Mathematics (CM.)

    To each entity from CM corresponds an infinity of similar entities in EM, therefore the

    Supermathematics (SM) is the reunion of the two domains: SM = CM EM, where CM is aparticular case of null ex-centricity ofEM. Namely, CM = SM(e = 0). To each known function in

    CM corresponds an infinite family of functions in EM, and in addition, a series of new functions

    appear, with a wide range of applications in mathematics and technology.In this way, to x = coscorresponds the family of functions x = cex = cex (, s, ) where s =

    e/R and are the polar coordinates of the ex-center S(s,), which corresponds to theunity/trigonometric circle orE(e, ),which corresponds to a certain circle of radius R, considered aspoleof a straight line d, which rotates around EorS with the position angle , generating in this waythe ex-centric trigonometric functions, or ex-centric circular supermathematics functions (EC-SMF),by intersecting d with the unity circle (see.Fig.1). Amongst them the ex-centriccosineof, denotedcex = x, where x is the projection of the point W, which is the intersection of the straight line withthe trigonometric circle C(1,O), or the Cartesian coordinates of the point W. Because a straight line,

    passing through S, interior to the circle (s 1e < R), intersects the circle in two points W1 and W2,which can be denoted W1,2, it results that there are two determinations of the ex-centric circular

    supermathematics functions (EC-SMF): a principal one of index 1cex1, and a secondary one cex2,of index 2, denoted cex1,2.EandS were named ex-centrebecause they were excluded from thecenter O(0,0). This exclusion leads to the apparition of EM and implicitly of SM. By this, the

    number of mathematical objects grew from one to infinity: to a unique function from CM, for

    example cos, corresponds aninfinity of functionscex, due to the possibilities of placing theex-center Sand/orE in the plane.

    S(e, ) can take an infinite number of positions in the plane containing the unity ortrigonometric circle. For each position ofS and E we obtain a functioncex. IfS is a fixed point,then we obtain the ex-centric circular SM functions (EC-SMF), with fixed ex-center, or with constant

    s and . But S or E can take different positions, in the plane, by various rules or laws, while the

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    8/133

    7

    straight line which generates the functions by its intersection with the circle, rotates with the angle aroundSandE.

    Fig.1 Definition of Ex-Centric Circular Supermathematics Functions (EC-SMF)

    In the last case, we have an EC-SMF of ex-center variable point S/E, which means s = s ()and/or = (). If the variable position ofS/E is represented also by EC-SMF of the same ex-centerS(s, ) or by another ex-center S1[s1 = s1(), 1 = 1 ()], then we obtain functions of double ex-centricity. By extrapolation, well obtain functions of triple, and multiple ex-centricity. Therefore,

    EC-SMFare functions of as many variables as we want or as many as we need .

    If the distances from O to the points W1,2on the circle C(1,O) areconstant and equal to the

    radius R = 1 of the trigonometric circle C, distances that will be named ex-centric radiuses, the

    distances from S to W1,2 denoted by r1,2 are variable and are named ex-centric radiuses of the unitycircle C(1,O) and represent, in the same time, new ex-centric circular supermathematics functions

    (EC-SMF), which were named ex-centric radial functions, denoted rex1,2 , if are expressed infunction of the variablenamed ex-centric and motor, which is the angle from the ex-centerE. Or,denoted Rex1,2, if it is expressed in function of the angle or the centric variable, the angle atO(0,0). The W1,2 are seen under the angles 1,2from O(0,0) and under the angles and + fromS(e, ) and E. The straight line d is divided by S din the twosemi-straight lines, one positived +and the other negative d

    . For this reason, we can considerr1 = rex1a positive oriented segment

    on d ( r1 > 0) andr2= rex2a negative orientedsegmenton d(r2 < 0) in the negative sense ofthe semi-straight line d

    .

    M1

    W1

    x

    y

    O

    S

    E

    cex1

    sex1

    cex2

    sex2

    OS = s OE = e

    OW1 = OW2 = 1OM1 = OM2 = R

    SW1 = r1 = rex1SW2 = r2 = rex2

    EM1 = R.r1 = R.rex1

    EM2 = R.r2 = R.rex2

    W1OA = 1W2OA = 2SOA = S(s,)E(e,)

    M1,2 (R, 1,2)

    W1,2 (1, 1,2)

    A

    aex1,2 = 1,2 () = 1,2() =

    bex1,2 == arcsin[s.sin(-)]cex1,2 = cos 1,2sex1,2 = sin 1,2

    W2

    dex1,2 =

    d

    d 2,1=

    =1 -

    )(sin1

    )cos(.

    22

    s

    s

    Dex 1,2 =

    21

    d

    d

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    9/133

    8

    Using simple trigonometric relations, in certain triangles OEW1,2, or, more precisely, writing

    the sine theorem (as function of) and Pitagoras generalized theorem (for the variables1,2) in thesetriangles, it immediately results the invariant expressions of the ex-centric radial functions:

    r1,2 () = rex1,2= s.cos() )(sin122 s

    and

    r1,2 (1,2) = Rex1,2 = )cos(..212 + ss .

    All EC-SMF have invariant expressions, and because of that they dont need to be tabulated,tabulated being only the centric functions from CM, which are used to express them. In all of their

    expressions, we will always find one of the square roots of the previous expressions, of ex-centric

    radial functions.Finding these two determinations is simple: for+ (plus) in front of the square roots we always

    obtain the first determination (r1 > 0) and for the(minus) sign we obtain the second determination(r2 < 0). The rule remains true for allEC-SMF. By convention, the first determination, of index 1, can

    be used or written without index.Some remarks about these REX (King) functions:

    The ex-centric radial functions are the expression of the distance between two points,in the plane, in polar coordinates: S(s, ) and W1,2 (R=1, 1,2), on the direction of thestraight line d, skewedat an angle in relation to Ox axis;

    Therefore, using exclusively these functions, we can express the equations of allknown plane curves, as well as of other new ones, which surfaced with theintroduction ofEM.An example is represented by Booths lemniscates (see Fig. 2, a,

    b, c), expressed, in polar coordinates, by the equation:

    () = R(rex1+rex 2) =2 s.Rcos( - ) forR=1, = 0 ands [0, 3].

    -1 -0.5 0.5 1

    -0.4

    -0.2

    0.2

    0.4

    -1 -0.5 0.5

    -0.2

    -0.1

    0.1

    0.2

    Fig.2,a Booths Lemniscates for R = 1 and

    numerical ex-centricity e [1.1, 2]Fig. 2,b BoothsLemniscates forR = 1and

    numerical ex-centricity e [2.1, 3]

    Another consequence is the generalization of the definition of a circle:The Circle is the plane curve whose points M are at the distances r() = R.rex =R.rex [, E(e, )] in relation to a certain point from the circles plane E(e, ).

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    10/133

    9

    IfSO(0,0), thens = 0 and rex = 1 = constant, andr() = R= constant, we obtainthe circles classical definition: the points situated at the same distance R from apoint, the center of the circle.

    Booth Lemniscate Functions

    Polar coordinate equation with

    supermathematics circle functions rex 1,2 : = R (rex1 + rex2 )for

    circle radius R = 1

    and

    the numerical ex-centricity s [ 0, 1 ]

    Fig. 2,c

    The functionsrex and Rex expresses the transfer functions of zero degree, or ofthe position of transfer, from the mechanism theory, and it is the ratio between the

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    11/133

    10

    parameter R(1,2), which positions the conducted element OM1,2 and parameterR.r1,2(), which positions the leader elementEM1,2.Between these two parameters, there are the following relations, which can be deduced

    similarly easy from Fig. 1 that defines EC-SMF.

    Between the position angles of the two elements, leaded and leader, there are the

    following relations:1,2 = arcsin[e.sin()] = 1,2() =aex1,2and

    = 1,2 1,2(1,2 )= 1,2 arcsin[)cos(..21

    )sin(.

    2,1

    2

    2,1

    +

    ss

    s] = Aex (1,2 ).

    The functions aex 1,2 and Aex 1,2 are EC-SMF, called ex-centric amplitude,because of their usage in defining the ex-centric cosine and sine from EC-SMF, in the

    same manner as the amplitude function or amplitudinus am(k,u) is used for defining

    the elliptical Jacobi functions:sn(k,u) = sn[am(k,u)], cn(k,u) = cos[am(k,u)],

    or:

    cex1,2 = cos(aex1,2) , Cex1,2 = cos(Aex1,2)and

    sex 1,2 = sin (aex1,2), Sex 1,2 = cos (Aex 1,2)

    The radial ex-centric functions can be considered as modules of the position vectors

    2,1r for the W1,2 on the unity circle C (1,O). These vectors are expressed by the

    following relations:

    radrexr .2,12,1 =

    ,

    where rad is the unity vector of variable direction, or the versor/phasor of thestraight line direction d

    +, whose derivative is the phasor der = d(rad)/d and

    represents normal vectors on the straight lines OW1,2, directions, tangent to the circle in

    the W1,2. They are named the centric derivative phasors. In the same time, themodulus rad function is the corresponding, in CM, of the function rex fors = 0 = when rex = 1 andder 1,2 are the tangent versors to the unity circle in W1,2.

    The derivative of the

    2,1r vectors are the velocity vectors:

    2,12,1

    2,1

    2,1 .

    derdexd

    rdv ==

    of the W1,2 C points in their rotating motion on the circle, with velocities of variablemodulus v1,2 =dex1,2, when the generating straight line d rotates around the ex-centerS with a constant angular speed and equal to the unity, namely = 1.The velocityvectors have the expressions presented above, where der 1,2are the phasors of centricradiuses R1,2 of module 1 and of1,2 directions. The expressions of the functions EC-

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    12/133

    11

    SMdex1,2 ,ex-centric derivativeof, are, in the same time, also the1,2() anglesderivatives, as function of the motor or independent variable , namely

    dex1,2 = d1,2 ()/d = 1)(sin.1

    )cos(.

    22

    s

    s

    as function of, and

    Dex 1,2 = d()/d1,2 =2,1

    2

    2,1

    2,1

    2

    2,1

    Re

    )cos(.1

    )cos(..21

    )cos(.1

    x

    s

    ss

    s =

    +

    ,

    as functions of 1,2 .It has been demonstrated that the ex-centric derivative functionsEC-SMexpress the

    transfer functions of the first order, or of the angular velocity, from the MechanismsTheory, forall (!) known plane mechanisms.

    The radial ex-centric function rex expresses exactly the movement of push-pullmechanism S = R. rex, whose motor connecting rod has the length r, equal with ethe real ex-centricity,and the length of the crank is equal to R, the radius of the circle,a very well-known mechanism, because it is a component of all automobiles, except

    those with Wankel engine.The applications of radial ex-centric functions could continue, but we will concentrate now on

    the more general applications ofEC-SMF.

    Concretely, to the unique forms as those of the circle, square, parabola, ellipse, hyperbola,

    different spirals, etc. from CM, which are now grouped under the name of centrics, correspond aninfinity ofex-centrics of the same type: circular, square (quadrilobe), parabolic, elliptic, hyperbolic,

    various spirals ex-centrics, etc. Any ex-centric function, with null ex-centricity (e = 0), degenerates

    into a centric function, which represents, at the same time its generating curve. Therefore, the CM

    itself belongs to EM, for the unique case (s = e = 0), which is one case from an infinity of possiblecases, in which a point named eccenterE(e, )can be placed in plane. In this case, E is overleapingon one or two points named center: the origin O(0,0) of a frame, considered the origin O(0,0) of the

    referential system, and/or the centerC(0,0) of the unity circle for circular functions, respectively, thesymmetry center of the two arms of the equilateral hyperbola, for hyperbolic functions.

    It was enough that a point E be eliminated from the center (O and/orC) to generate from the

    old CM a new world ofEM. The reunion of these two worlds gave birth to theSM world.This discovery occurred in the city of the Romanian Revolution from 1989, Timioara, which

    is the same citywhere on November 3rd

    , 1823 Janos Bolyai wrote: FromnothingIve created a new

    world. With these words, he announced the discovery of the fundamental formula of the first non-

    Euclidean geometry.

    He from nothing, I in a joint effort, proliferated the periodical functions which are so

    helpful to engineers to describe some periodical phenomena. In this way, I have enriched the

    mathematics with new objects.

    When Euler defined the trigonometric functions, as direct circular functions, if he wouldnthave chosen three superposed points: the origin O, the center of the circle C and S as a pole of a

    semi straight line, with which he intersected the trigonometric/unity circle, the EC-SMF would havebeen discovered much earlier, eventually under another name.

    Depending on the way of the split (we isolate one point at the time from the superposed

    ones, or all of them at once), we obtain the following types ofSMF:

    O C S Centric functions belonging to CM;

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    13/133

    12

    and those which belong toEM are:

    O C S Ex-centric Circular SupermathematicsFunctions (EC-SMF);O C S Elevated Circular SupermathematicsFunctions (ELC-SMF);O C S Exotic Circular SupermathematicsFunctions (EXC-SMF).These new mathematics complements, joined under the temporary name of SM, are

    extremely useful tools or instruments, long awaited for. The proof is in the large number and thediversity of periodical functions introduced in mathematics, and, sometimes, the complex way of

    reaching them, by trying the substitution of the circle with other curves, most of them closed.

    To obtain new special, periodical functions, it has been attempted the replacement of the

    trigonometric circle with the square or the diamond. This was the proceeding of Prof. Dr. Math.

    Valeriu Alaci, the former head of the Mathematics Department of Mechanics College from

    Timioara, who discovered the square and diamond trigonometric functions. Hereafter, themathematics teacherEugenVisa introduced the pseudo-hyperbolic functions, and the mathematicsteacher M. O. Enculescu defined the polygonal functions, replacing the circle with an n-sides

    polygon; for n = 4 he obtained the square Alaci trigonometric functions. Recently, the mathematician,Prof. Malvina Baica, (of Romanian origin) from the University of Wisconsin together with Prof.

    Mircea Crdu, have completed the gap between the Euler circular functions and Alaci squarefunctions, with the so-called Periodic Transtrigonometric functios.

    The Russian mathematician Marcusevici describes, in his work Remarcable sine functions

    the generalized trigonometric functions and the trigonometric functions lemniscates.

    Even since 1877, the German mathematician Dr. Biehringer, substituting the right trianglewith an oblique triangle, has defined the inclined trigonometric functions. The British scientist of

    Romanian origin EngineerGeorge (Gogu) Constantinescu replaced the circle with the evolventand

    defined the Romaniantrigonometric functions:Romanian cosine and Romanian sine, expressed by

    Corand Sirfunctions,which helped him to resolve some non-linear differential equations of theSonicity Theory, which he created. And how little known are all these functions even in Romania!

    Also the elliptical functions are defined on an ellipse. A rotated one, with its main axis along

    Oy axis.How simple the complicated things can become, and as a matter of fact they are! This

    paradox(ism) suggests that by a simple displacement/expulsion of a point from a center and by the

    apparition of the notion of the ex-center, a new world appeared, the world of EMand, at the sametime, a new Universe, the SMUniverse.

    Notions like Supermathematics Functions and Circular Ex-centric Functions

    appeared on most search engines like Google, Yahoo, AltaVista etc., from the beginning of the

    Internet. The new notions, like quadrilobequadrilobas, how the ex-centrics are named, and whichcontinuously fill the space between a square circumscribed to a circle and the circle itself were

    included in the Mathematics Dictionary. The intersection of the quadriloba with the straight line dgenerates the new functions called cosinequadrilobe-ic andsine quadrilobe-ic.

    The benefits ofSM in science and technology are too numerous to list them all here. But we

    are pleased to remark that SM removes the boundaries between linear and non-linear; the linear

    belongs to CM, and the non-linear is the appanage ofEM, as between ideal and real, or as betweenperfection and imperfection.

    It is known that the Topology does not differentiate between a pretzel and a cup of tea.Well,

    SM does not differentiate between a circle(e = 0) and a perfect square (s = 1), between a circle

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    14/133

    13

    and a perfect triangle, between an ellipse and a perfect rectangle, between a sphere and a perfectcube, etc. With the same parametric equations we can obtain, besides the ideal forms ofCM (circle,

    ellipse, sphere etc.), also thereal ones (square, oblong, cube, etc.). Fors [-1,1], in the case of ex-centric functions of variable, as in the case of centric functions of variable , fors[-,+], it can be obtained an infinity of intermediate forms, for example, square, oblong or cube with rounded

    corners and slightly curved sides or, respectively, faces. All of these facilitate the utilization of thenew SM functions for drawing and representing of some technical parts, with rounded or splayed

    edges, in the CAD/ CAM-SM programs, which dont use the computer as drawing boards any more,

    but create the technical object instantly, by using the parametric equations, that speed up theprocessing, because only the equations are memorized, not the vast number of pixels which define the

    technical piece.

    The numerous functions presented here, are introduced in mathematics for the first time,

    therefore, for a better understanding, the author considered that it was necessary to have a short presentation of their equations, such that the readers, who wish to use them in their applications

    development, be able to do it.

    SM is not a finished work; its merely an introduction in this vast domain, a first step, the

    authors small step, and a giant leap for mathematics.The elevated circular SM functions (ELC-SMF), named this way because by the

    modification of the numerical ex-centricity s the points of the curves of elevated sine functions sel as of the elevated circular function elevated cosine celis elevating in other words it rises on thevertical, getting out from the space {-1, +1] of the other sine and cosine functions, centric or ex-

    centric. The functions cexandsex plots are shownin Fig. 3, where it can be seen that the pointsof these graphs get modified on the horizontal direction, but all remaining in the space

    [-1,+1], named the existence domain of these functions.

    The functions cel and sel plots can be simply represented by the products:

    cel 1,2 = rex1,2 . cos and Cel1,2= Rex 1,2. cos

    sel 1,2 = rex 1,2 . sin and Sel 1,2 = Rex 1,2. sin

    and are shown Fig. 4.

    The exotic circular functionsare the most general SM, and are defined on the unity circle

    which is not centered in the origin of the xOy axis system, neither in the eccenterS,but in a certain

    point C (c,) from the plane of the unity circle, of polar coordinates (c, ) in the xOy coordinatesystem.

    Many of the drawings from this album are done with EC-SMF of ex-center variable and with

    arcs that are multiples of (n.). The used relations for each particular case are explicitly shown, inmost cases using the centric mathematical functions, with which, as we saw, we could express all SM

    functions, especially when the image programs cannot use SMF. This doesnt mean that, in the future,the new math complements will not be implemented in computers, to facilitate their vast utilization.

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    15/133

    14

    Fig. 3,a The ex-centric circular

    supermathematicsfunction

    (EC-SMF) ex-centric cosine ofcex

    for = 0, [0, 2]

    Fig. 3,b The ex-centric circular

    supermathematicsfunction

    (EC-SMF) ecentric sine ofsex for

    = 0, [0, 2]Numerical ex-centricity s = e/R [ -1, 1]

    The computer specialists working in programming the computer assisted design software

    CAD/CAM/CAE, are on their way to develop these new programs fundamentally different, becausethe technical objects are created with parametric circular or hyperbolic SMFs, as it has been

    exemplified already with some achievements such as airplanes, buildings, etc. in

    http://www.eng.upt.ro/~mselariuand how a washer can be represented as a toroid ex-centricity (or asan ex-centric torus), square or oblong in an axial section, and, respectively, a square plate with a

    central square hole can be a square torus of square section. And all of these, because SMdoesnt

    make distinction between a circle and a square or between an ellipse and a rectangle, as we mentionedbefore.

    But the most important achievements in science can be obtained by solving some non-linearproblems, because SMreunites these two domains, so different in the past, in a single entity. Among

    these differences we mention that the non-linear domain asks for ingenious approaches for each

    problem. For example, in the domain of vibrations, static elastic characteristics (SEC) soft non-linear

    (regressive) or hard non-linear (progressive) can be obtained simply by writing y = m. x, where m is

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    16/133

    15

    not anymorem = tan as in the linear case (s = 0 ), but m = tex1,2 and depending on the numericalex-centricity s sign, positive or negative, or for S placed on the negative x axis ( =) or on thepositive x axis ( = 0), we obtain the two nonlinear elastic characteristics, and obviously for s=0 wellobtain the linear SEC.

    1 2 3 4 5 6

    -1

    -0.5

    0.5

    1

    1.5

    2

    1 2 3 4 5 6

    -2

    -1.5

    -1

    -0.5

    0.5

    1

    1 2 3 4 5 6

    -1

    -0.5

    0.5

    1

    1.5

    1 2 3 4 5 6

    -1.5

    -1

    -0.5

    0.5

    1

    Fig. 4,a ELC-SMF elevated cosine of - cel, fors [-1, +1], = 0, [0, 2].

    Fig. 4,b ELC-SMFelevated sine of - sel, fors[-1, +1], = 1, [0, 2].

    Due to the fact that the functions cex and sex , as well Cex and Sex and theircombinations, are solutions of some differential equations of second degree with variable coefficients,it has been stated that thelinear systems (Tchebychev) are obtained also fors = 1, and not onlyfors

    = 0. In these equations, the mass ( the point M) rotates on the circle with a double angular speed =2. (reported to the linear system where s = 0 and = = constant) in a half of a period, and inthe other half of period stops in the point A(R,0) fore = sR= R or = 0and in A(R, 0) for e = s.R= 1, or = . Therefore, the oscillation period T of the three linear systems is the same andequal with T=/ 2. The nonlinear SEC systems are obtained for the others values, intermediates,ofs and e. The projection, on any direction, of the rotating motion of M on the circlewith radius R,

    equal to the oscillation amplitude, of a variable angular speed = .dex ( afterdex function) isan non-linear oscillating motion.

    The discovery of king function rex, with its properties, facilitated the apparition of ahybrid method (analytic-numerical), by which a simple relation was obtained, with only two terms,

    to calculate the first degree elliptic complete integral K(k), with an unbelievable precision, with a

    minimum of15 accurate decimals, after only 5 steps. Continuing with the next steps, can lead us to anew relation to compute K(k), with a considerable higher precision and with possibilities to expand

    the method to other elliptic integrals, and not only to those. After 6 steps, the relation ofE (k) has the

    same precision of computation.

    s [0, 1]

    s [ -1, 0]

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    17/133

    16

    The discovery of SMF facilitated the apparition of a new integration method, named

    integration through the differential dividing.We will stop here, letting to the readers the pleasure to delight themselves by viewing the

    drawings from this album.

    [Translated from Romanian by Marian Niu and Florentin Smarandache]

    Mircea Eugen elariu

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    18/133

    17

    Selariu SuperMathematics Functions

    &

    Other SuperMathematics Functions

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    19/133

    18

    D O U B L E C L E P S Y D R A

    -0.5

    0

    0.5

    -0.5

    0

    0.5

    -1

    -0.5

    0

    0.5

    1

    -0.5

    0

    0.5

    M

    =

    =

    =

    sexz

    ucexy

    ucexx

    sin.

    cos.

    , EccenterS(1,0) s = 1, = 0, t [0, 3]; u [0, 2]

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    20/133

    19

    S U P E R M A T H E M A T I C S F L O W E R S

    0.5 1 1.5 2

    0.5

    1

    1.5

    2

    0.5 1 1.5 2

    0.5

    1

    1.5

    2

    X = dex (/`2), s = s0 cos 4 , = 0

    Y = dex , s0 [0, 1], [ 0, 2 ]X = dex ( + /`2), s = s0 cos 6 , = 0

    Y = dex , s0 [0, 1], [ 0, 2 ]

    0.5 1 1.5 2

    0.5

    1

    1.5

    2

    0.5 1 1.5 2

    0.5

    1

    1.5

    2

    X = dex ( + /`2), s = s0 cos 8 , = 0

    Y = dex , s0 [0, 1], [ 0, 2 ]

    X = dex ( + /`2), s = s0 cos 15 , = 0

    Y = dex , s0 [0, 1], [ 0, 2 ]

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    21/133

    20

    The Ballet of the Functions

    0 1 2 3 4 5 6

    -1

    0

    1

    2

    0 1 2 3 4 5 6

    -0.4

    -0.2

    0

    0.2

    0.4

    0.6

    F(x) = sign(cos x).cos x / (1+ tann x)n, , n [0, 10] F (x) = sign(sin x).sin x / (1+ tann x)n, , n [0, 10]

    0 1 2 3 4 5 6

    -0.2

    -0.1

    0

    0.1

    0 1 2 3 4 5 6

    -0.2

    -0.1

    0

    0.1

    0.2

    F (x) = sign(cos x) tan x (1 + Abs(tan n x))n, n [0, 10] F (x) = sign(sin x) tan x (1 + Abs(tan n x))n, n [0, 10]

    0 1 2 3 4 5 6

    -0.15

    -0.1

    -0.05

    0

    0.05

    0.1

    0.15

    0 1 2 3 4 5 6

    -0.15

    -0.1

    -0.05

    0

    0.05

    0.1

    0.15

    F (x) = sign(sin x) tan x / (1 + Abs(tan x))n, n [0, 10] F (x) = sign(cos x) tan x / (1 + Abs(tan x))n, n [0, 10]

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    22/133

    21

    J a c u z z i

    -2

    -1

    0

    1

    2-2

    -1

    0

    1

    2

    -1

    -0.5

    0

    0.5

    1

    -2

    -1

    0

    1

    2

    -2

    -1

    0

    1

    2

    -2

    -1

    0

    1

    2

    -1

    -0.5

    0

    0.5

    1

    -2

    -1

    0

    1

    2

    M

    ==

    =

    ==

    ]3,0[....1,.

    ]2,0[,..sin).1(

    2sin,..cos).1(

    ssexz

    uucexy

    usucexx

    M

    ==

    ==

    ==

    ]3,0[.....................1,.

    ]2,0[,cos,.sin).1(

    sin........,.........cos).1(

    ssexz

    uusucexy

    usucexx

    I m p l o d e d J a c u z z i-2

    -1

    0

    1

    2

    -0.5

    0

    0.5

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    M

    ==

    ==

    ==

    ]3,0[.....................1,.

    ]2,0[,sin,.sin).1(

    sin........,.........cos).1(

    ssexz

    uusucexy

    usucexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    23/133

    22

    D a m a g e d p a r t o f T I T A N I C

    -0.5

    0

    0.5

    -2

    -1

    0

    1

    2

    -1

    -0.5

    0

    0.5

    1

    -0.5

    0

    0.5

    -2

    -1

    0

    1

    2

    M } ]2,0[],3,0[,1)cos.1(cos).1(

    =

    =

    +=

    =

    us

    sexz

    ucexy

    ucexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    24/133

    23

    K A Z A T C I O K (Russian Popular Dance)

    -2 -1 1 2

    -2

    -1

    1

    2

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    25/133

    24

    F l y i n g B i r d 1

    -2 -1 1

    -2

    -1

    1

    2

    M

    +===

    ==

    3cos)8.0',0(.5sin1

    3cos])1,0[,0,(.5sin12

    2

    sSsexy

    scexx, S=s.cos5, ]2,0[ ,s ]1,0[

    -1 -0.5 0.5 1 1.5

    -1

    -0.5

    0.5

    1

    1.5

    M ]2,0[],1,0[,5cos.)5cos(.)((sin1)8.0,(

    )5cos(.)3(sin1),(2

    2

    =

    +==

    +=ssS

    SsSssexy

    SsSscexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    26/133

    25

    F l y i n g B i r d 2

    -1 -0.5 0.5 1 1.5

    -2

    -1

    1

    2

    M ]2,0[,0],1,0[,3cos9sin1)8.0,(

    3cos.5cos.3sin1),(2

    2

    =

    ++==

    +=s

    ssexy

    sscexx

    -1 -0.5 0.5 1 1.5

    -1.5

    -1

    -0.5

    0.5

    1

    1.5

    M ]2,0[,0],1,0[,5cos.3cos.9sin1)8.0,(

    3cos.5cos.3sin1),(2

    2

    =

    ++==

    +=s

    sssexy

    sscexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    27/133

    26

    M U L T I C O L O R E D S U N

    -2 -1 1 2

    -2

    -1

    1

    2

    M ]2,0[],1,1[,sin..20

    cos..20

    =

    =

    sdexy

    dexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    28/133

    27

    R e d S u n

    -2 -1 1 2

    -2

    -1

    1

    2

    M ]2,0[],1,1[,sin..20

    cos..20

    =

    =s

    dexy

    dexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    29/133

    28

    T h e d o u b l e N o z z l e f o r N A S A

    -2-1

    01

    2

    -2

    -1

    0

    1

    2

    -2

    0

    2

    -2

    -1

    0

    1

    2

    -2-1

    01

    2

    -1

    0

    1

    -2

    0

    2

    -1

    0

    1

    -2-1

    01

    2

    -2

    -1

    0

    1

    2

    -2

    0

    2

    -2

    -1

    0

    1

    2

    --1

    01

    2

    -2

    -1

    0

    1

    -2

    0

    2

    -2

    -1

    0

    1

    M

    =

    =

    =

    sz

    xny

    xnx

    3

    sin..Re

    cos..Re

    , s [-1, 1], [0, 2], n = 2 ,3 ,4, 5.

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    30/133

    29

    T h e s u p e r m a t h e m a t i c s C o m e t

    -4 -3 -2 -1 1

    -2

    -1

    1

    2

    M ]2,0[),0],1,0[(,sin.

    cos.

    =

    =

    =sS

    dexy

    dexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    31/133

    30

    The Lake of Swans The Dance of Swords

    0 1 2 3 4 5 6

    -1

    0

    1

    2

    0 1 2 3 4 5 6

    -1.5

    -1

    -0.5

    0

    0.5

    1

    1.5

    nnx

    xxsign

    )tan1(

    cos)(cos

    +, n [0, 10]; x [0, 2]

    x

    xxsignntan1

    cos)(

    +, n [-10, 10]; x [0, 2]

    The Nut Cracker The Decease of Swan

    0 1 2 3 4 5 6

    -1

    -0.5

    0

    0.5

    1

    0 1 2 3 4 5 6

    -0.4

    -0.2

    0

    0.2

    0.4

    0.6

    x

    xxsignn 2tan1

    5cos)(sin

    + , n [-10, 10]; x [0, 2] nnx

    xxsign

    )tan1(

    sin)(sin

    + , n [0, 10]; x [0, 2]

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    32/133

    31

    T h e F l o w e r i n g

    0.6 0.8 1.2 1.4

    0.6

    0.8

    1.2

    1.4

    0.4 0.6 0.8 1.2 1.4 1.6

    0.6

    0.8

    1.2

    1.4

    M

    ==

    ==

    2,0[,,4cos.,

    ,4sin.,4cos.),

    1

    21

    ssdexy

    sssdexx

    s [0, 1], = - /2, [0, 2,]

    M

    ==

    ==

    2,0[,,8cos.,

    ,8sin.,8cos.),

    1

    21

    ssdexy

    sssdexx

    s [0, 1], = - /2, [0, 2,]

    0.5 1 1.5 2

    0.6

    0.8

    1.2

    1.4

    0.4 0.6 0.8 1.2 1.4 1.6

    0.6

    0.8

    1.2

    1.4

    M

    ====

    ====

    0,8cos.,2cos.,

    2/,2sin.,2cos.,2/3

    2

    3

    1

    2

    2

    1

    ssssdexy

    ssssdexx

    s [0, 1], [0, 2]

    ====

    ====

    0,2cos.,2cos.,

    2/,2sin.,2cos.,2/3

    2

    3

    1

    2

    2

    1

    ssssdexy

    ssssdexx

    s [0, 1], [0, 2]

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    33/133

    32

    The supermathematics ring surface

    -4

    -2

    0

    2

    4-4

    -2

    0

    2

    4

    -1-0.5

    00.5

    1

    -4

    -2

    0

    2

    4

    -4

    -2

    0

    2

    4-4

    -2

    0

    2

    4

    -1

    -0.5

    0

    0.5

    1

    -4

    -2

    0

    2

    4

    M

    +=

    +=

    q

    quqy

    quqx

    sin

    sin).cos3(

    cos).cos3(

    s =1, = 0, [0, 2], u [, ]

    -2

    0

    2

    -2

    0

    2

    -1

    -0.5

    0

    0.5

    1

    -2

    0

    2

    -2

    0

    2-2

    0

    2

    -1

    -0.5

    0

    0.5

    1

    -2

    0

    2

    M

    +=

    +=

    sex

    ucexy

    ucexx

    sin).2(

    cos).2(

    s =1, = 0, [0, 2], u [0, 2]

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    34/133

    33

    The ex-centric sphere

    -2

    -1

    0

    1

    2

    -1

    0

    1-1

    -0.5

    0

    0.5

    1

    -2

    -1

    0

    1

    2

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    M

    =

    =

    =

    sin.

    sin.cos.

    cos.cos.

    dexz

    udexy

    udexx

    S(s [0,1], =0) M

    =

    =

    =

    sin.cos

    sin.cos.sin

    cos.cos.cos

    qz

    uqy

    uqx

    S(s [0,1], =0)

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    M

    =

    ==

    uz

    usexy

    ucexx

    sin

    sin.

    cos.

    S(s [0,1], =0) M

    =

    ==

    uqz

    uqy

    uqx

    sin.cos

    sin.sin

    cos.cos

    S(s [0,1], =0)

    [0, 2], u [, ], S[ s [ 0, 1], = 0]

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    35/133

    34

    T h e s u p e r m a t h e m a t i c s Screw Surface

    -20

    -10

    0

    10

    20

    -20

    -10

    0

    10

    -20

    0

    20

    40

    60

    -20

    -10

    0

    10

    M

    ++===

    ++=

    ++=

    )13

    2sin(

    13

    8.4)0,1,

    4(.8

    )]13

    2cos(5sin[

    13

    2

    )]13

    2cos(5cos[

    13

    2

    ussexz

    uy

    ux

    , u [0, 2], [0, 26]

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    36/133

    35

    T h e T r o j a n H o r s e

    0

    2

    4

    6

    -1

    -0.5

    0

    0.5

    1

    -4

    -2

    0

    2

    4

    -1

    -0.5

    0

    0.5

    1

    -4

    -2

    0

    2

    4

    0

    2

    4

    6

    -1-0.5

    00.5

    1

    -4

    -2

    0

    2

    4

    -1-0.5

    00.5

    1

    -4

    -2

    0

    2

    4

    X = , y = s, Z = cosq [, S(s, )],

    S[s

    [ - 1, 1], = 0],

    [0, 2]

    X = , y = s, Z = sinq [, S(s, )],

    S[s

    [ - 1, 1], = 0],

    [0, 2]

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    37/133

    36

    T h e A m p h o r a s

    -2

    -1

    0

    12

    -1

    0

    1

    -10

    -7.5

    -5

    -2.5

    0

    -1

    0

    1

    -1

    0

    1

    -1

    0

    1

    0

    2

    4

    6

    -1

    0

    1

    M

    =

    =+=

    =+=

    ]2,0[],5.0,6.3[,...9.0

    sin.)2

    cos,(.21

    cos.)98.0,(.21

    2

    2

    z

    ssexssy

    scexssx

    M ]2,0[,

    ]2.2,0[

    sin.Re

    cos.Re

    =

    =

    =

    sz

    xy

    xx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    38/133

    37

    B U D D H A

    -4

    -2

    0

    2

    4-2

    -1

    0

    1

    2

    -10

    -7.5

    -5

    -2.5

    0

    -2

    -1

    0

    1

    2

    M

    =

    =+=

    =+=

    ]2,0[],5.0,6.3[,9.0

    sin.)cos,(..214.1

    cos.)98.0,(.212

    2

    2

    1

    2

    ssz

    sscexssy

    sssexssx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    39/133

    38

    J E T P L A N E

    0

    2

    4

    6

    8-2

    0

    2

    0

    1

    2

    0

    2

    4

    6

    8

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    40/133

    39

    T R O U B L E D L A N Do r

    D O U B L E A N A L Y N I C A L E X C E N T R I C F U N C T I O N

    0

    5

    101

    2

    3

    4

    0

    0.5

    1

    0

    5

    10

    cex2a(x,S(s, = 0), ) = cos{ )]2

    arcsin(2

    [2

    xxx }

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    41/133

    40

    S E L F - P I E R C E B O D Y

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1-0.5 0 0.5 1

    -1

    -0.5

    0

    0.5

    1-1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1 -0.5 0

    0.5 1

    -1

    -0.5

    0

    0.5

    1-1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.51

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.51

    -1

    -0.5

    0

    0.5

    1

    M

    =

    =

    ==

    ])2,0[(.2sin

    ])2,0[(.2sin

    )9.0,3(

    scexz

    ssexy

    sbexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    42/133

    41

    H I L L S a n d V A L L E Y S

    - 2

    0

    2

    - 2

    0

    2

    - 8

    - 6

    - 4

    - 2

    0

    - 2

    0

    2

    z = 10/(1 + 1.3 x2

    + y2)[sex[(x

    2- y

    2), S(s = 0.8, = 0)] Rex[(x2 y2), S(s = 0.64, = 0)],

    x, y [ - , + ]

    -2

    0

    2

    0

    1

    2

    3

    -6

    -4

    -2

    0

    -2

    0

    2

    z = 10/(1 + 1.3 x2 + y2)[sex[(x2 - y2 - 8), S(s = 0.8, = 0)] Rex[(x2 y2 - 8), S(s = 0.64, = 0)],x [ - , + ], y [ 0, + ],

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    43/133

    42

    B e r n o u l l i s L e m n i s c a t e, C a s s i n i s O v a l s a n d o t h e r s

    0.4 0.6 0.8 1.2 1.4 1.6

    0.5

    1

    1.5

    2

    M ]2,0[),2

    ],1,0[(2cos.,2(

    )cos,(0

    0

    0

    =

    ==

    ==sS

    ssdexy

    ssdexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    44/133

    43

    Continuous transforming of a circle into a haystack

    0.4 0.6 0.8 1.2 1.4 1.6

    0.2

    0.4

    0.6

    0.8

    1

    1.2 1.4 1.6 1.8 2

    0.4

    0.6

    0.8

    1.2

    1.4

    1.6

    M ]2,0[],1,0[

    ))0,cos(,(

    ))2

    ,cos)(,(0

    0

    0

    ===

    ===s

    ssSdexy

    ssSdexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    45/133

    44

    C y c l i c a l S y m m e t r y

    0.4 0.6 0.8 1.2 1.4 1.6

    0.25

    0.5

    0.75

    1

    1.25

    1.5

    1.75

    0 . 4 0 . 6 0 . 8 1 . 2 1 . 4 1 . 6

    0 . 2 5

    0. 5

    0 . 7 5

    1

    1 . 2 5

    1. 5

    1 . 7 5

    0.5 1 1.5 2

    0.5

    1

    1.5

    2

    M M ]2,0[],1,0[

    ))0,4cos(,(

    ))2

    ,4cos)(,(0

    0

    0

    ===

    ===s

    ssSdexy

    ssSdexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    46/133

    45

    S m a r a n d a c h e S t e p p e d F u n c t i o n s

    2 4 6 8 10 12

    2.5

    5

    7.5

    10

    12.5

    15

    6 8 10 12

    -10

    -5

    5

    10

    F(n, s, ) = Ssf () = [ bex(, s = 1) dex ].s.dex n, n = 10, s = [0.2, 0.4, 0.6, 0.9, 1]

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    47/133

    46

    S C R I B B L I N G S W I T H . . . H E A D A N D T A L E

    -4 -2 2 4

    -4

    -2

    2

    4

    -4 -2 2 4

    -4

    -2

    2

    4

    M ]2,0[],1,0[,32)1(

    32)1(

    =

    +=s

    sexsexy

    cexcexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    48/133

    47

    Q U A D R I P O D

    -2

    0

    2

    -2

    0

    2

    -1

    -0.5

    0

    0.5

    1

    -2

    0

    2

    z = cexq [x.y, S(s = 0.8, = 0)], x, y [, + ]

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    49/133

    48

    E X C E N T R I C S Y M M E T R Y

    -4 -2 2 4

    -4

    -2

    2

    4

    -4 -2 2 4

    -4

    -2

    2

    4

    M ]2,0[),0],1,0[]0,1[()53sin(2.3

    72.3

    =+

    =

    +=andssS

    bexsexy

    cexcexx

    -4 -2 2 4

    -4

    -2

    2

    4

    M ]2,0[),0],1,1[()53sin(2.3

    72.3

    =+

    =

    +=sS

    bexsexy

    cexcexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    50/133

    49

    D R A C U L A S C A S T L E

    -2

    0

    2

    -2

    0

    2

    -1

    -0.5

    0

    0.5

    1

    -2

    0

    2

    -1

    0

    1

    -2

    0

    2

    0

    2.5

    5

    7.5

    10

    -1

    0

    1

    Z = cosq (x.y), s = 0.8,

    x, y [, + ]Z = 1 / cosq (x.y). cos (x.y), s = 0.8,

    x, y [, + ]

    -2

    0

    2

    -2

    0

    2

    -4

    -2

    0

    2

    4

    -2

    0

    2

    Z = 1 / cosq (x.y), s = 0.8, x,y [, + ]

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    51/133

    50

    T U N I N G F O R K

    -4

    -2

    0

    2

    4

    -10

    0

    10

    -1-0.5

    00.5

    1

    -4

    -2

    0

    2

    4

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    52/133

    51

    d e x - O I D S

    01

    2

    3

    -1-0.5

    0

    0.51

    0

    0.5

    1

    1.5

    2

    01

    2

    3

    0

    1 2

    3

    -

    -0.5

    00.5

    1

    0

    0.5

    1

    1.5

    2

    0

    1 2

    3

    z = dex(x, y), x [0, ], S( s y [-1,1], = 0)

    r e x O I D S

    0

    1

    23

    - 1

    - 0 . 5

    00 . 5

    1

    0

    0 . 5

    1

    1 . 5

    2

    0

    1

    23

    z = Rex(x, y), x [0, ], S( s y [-1,1], = 0)

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    53/133

    52

    E x c e n t r i c g e o m e t r y , p r i s m a t i c s o l i d s

    0

    0.5

    1

    1.5

    2

    0

    0.5

    1

    1.5

    2

    0

    0.5

    1

    1.5

    0

    0.5

    1

    1.5

    2

    0

    0.5

    1

    1.5

    2

    0

    0.5

    1

    1.5

    2

    0

    0.5

    1

    1.5

    -1.5

    -1

    -0.5

    0

    0

    0.5

    1

    1.5

    2

    0

    0.5

    1

    1.5

    M ]0,1[..]..1,0[],2,0[,

    5.1

    )2/,4(

    )0,4(

    =

    ==

    ==

    sors

    sz

    dexy

    dexx

    -1-0.5

    0

    0.51

    -1

    -0.5

    0

    0.5

    1

    -3

    -2

    -1

    0

    1

    -1

    -0.5

    0

    0.5

    1

    -1-0.5

    00.5

    1

    -1

    -0.5

    0

    0.5

    1

    -3

    -2

    -1

    0

    1

    -1

    -0.5

    0

    0.5

    1

    M1

    =

    =

    =

    2

    sin

    cos.

    1

    1

    1

    sz

    qy

    qsx

    , M2

    =

    =

    =

    sz

    qy

    qx

    sin

    cos

    2

    2

    , [0, 2], S( = 0, s [-1,1])

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    54/133

    53

    V a s e

    -1

    01

    -1

    0

    1

    -10

    -7.5

    -5

    -2.5

    0

    -1

    0

    1

    -2-1

    01

    2

    -1

    0

    1

    -10

    -7.5

    -5

    -2.5

    0

    -1

    0

    1

    M ]2,0[,

    ]0,6.3[,

    sin).(Recos).(Re

    =

    ==

    sz

    sxysxx

    M

    =

    =+==

    =+==

    ]2,0[],2/,6.3[,.9.0

    )5cos,(.21sin),(Resin

    ]98.0,[.21cos),(Recos

    222

    1

    2

    1

    ssz

    sssexsssxy

    sscexsssxx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    55/133

    54

    H E X A G O N A L T O R U S

    -4

    -2

    0

    2

    4-4

    -2

    0

    2

    4

    -1

    -0.5

    0

    0.5

    1

    -4

    -2

    0

    2

    4

    -4

    -2

    0

    2

    4-4

    -2

    0

    2

    4

    -1

    -0.5

    0

    0.5

    1

    -4

    -2

    0

    2

    4

    M ]2,0[],2,0[,

    ]0,1(,[

    sin]).0,1(,[3(

    cos]).0,1(,[3(

    111

    111

    111

    ===

    ==+=

    ==+=

    s

    sSssexz

    sSscexy

    sSscexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    56/133

    55

    O p e n s q u a r e t o r u s M ]2.2,0[],2,0[,)sin)cos3(

    cos).cos3(

    =

    +=

    +=

    s

    sz

    qsy

    qsx

    -4

    -2

    0

    2

    4

    -4

    -2

    0

    2

    4

    0

    2

    4

    6

    -4

    -2

    0

    2

    4

    -4

    -2

    0

    2

    4

    S q u a r e t o r u s M ]2.2,0[],2,0[,

    sin

    )1,(sin)cos3(

    )1,(cos).cos3(

    1

    1

    =

    =+=

    =+=

    s

    sz

    sqsy

    sqsx

    -4

    -2

    0

    2

    4-4

    -2

    0

    2

    4

    -1

    -0.5

    0

    0.5

    1

    -4

    -2

    0

    2

    4

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    57/133

    56

    D o u b l e s q u a r e t o r u s ]2,0[,,

    )1,(sin

    sin).cos1/)1,(cos3(

    cos).sin1/)1,(cos3(2

    2

    =

    +=

    +=

    s

    qz

    sqy

    sqx

    -4

    -2

    0

    2

    4-4

    -2

    0

    2

    4

    -1

    -0.5

    0

    0.5

    1

    -4

    -2

    0

    2

    4

    -2

    0

    2

    -2

    0

    2

    -1

    -0.5

    0

    0.5

    1

    -2

    0

    2

    S q u a r e t o r u s ]2,0[,

    )1,(sin

    sin)].1,(cos3[

    cos)].1,(cos2[

    =

    +=

    +=

    s

    sqz

    sqy

    sqx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    58/133

    57

    S i n u o u s C o r r u g a t e W a s h e r s

    o r

    N a n o - P e r i s t a l t i c E n g i n e

    -4

    -2

    0

    2

    4

    -4

    -2

    0

    2

    4

    0

    1

    2

    3

    -4

    -2

    0

    2

    4

    M

    +

    ++=

    +=

    +=

    30)

    3/208sin(2.0)1,(sin3.0

    ]sin).1,(cos3[

    ]cos)].1,(cos3[

    sqz

    sqy

    sqx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    59/133

    58

    I G L O O M]2,0[,,

    1.0

    ),(sin4

    cos),(cos4

    sin.22

    ),(sin4

    sin),(cos4

    cos.22

    4

    =

    =

    =

    s

    sz

    sqsqsy

    sqsqsx

    -1

    0

    1

    -1

    0

    1

    0

    0. 5

    1

    1. 5

    -1

    0

    1

    T h e m a g i c c a r p e t ]1,1[],2,0[)),0,(,( 2 === syxysSxbexz

    0

    2

    4

    6 -1

    - 0 . 5

    0

    0. 5

    1

    - 0 . 5

    0

    0. 5

    0

    2

    4

    6

    - 0 . 5

    0

    0. 5

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    60/133

    59

    Multiple Ex Centric Circular SuperMathematics Functions

    1 2 3 4 5 6

    -1

    - 0 . 5

    0. 5

    1

    sex with dauble ex centre: y = s2ex =sin[-arcsin[s.sin]-arcsin[s.sin[-arcsin[s.sin ]]]]

    ex centre S( s [-1, 1], = 0 ), [0, 2]

    1 2 3 4 5 6

    -1

    -0.5

    0. 5

    1

    y = sin[ arctan[s.sin2 /Rex 2 ]]]

    1 2 3 4 5 6

    - 1

    - 0 . 5

    0 . 5

    1

    y = cos[ arctan(s.sin2)] / 2cos.sa

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    61/133

    60

    E X C E N T R I C T O R U S R I N G

    M

    ===+=

    =+=

    ],0[]2,0[],2,0[,

    )8.0,(

    sin)]}sin.9.0,2(cos[2{

    cos)]}8.0,2(cos[2{

    u

    ssexz

    uusbexy

    usbexx

    -2

    0

    2

    -2

    0

    2

    -1-0.5

    0

    0.5

    1

    -2

    0

    2

    -2

    0

    20

    1

    2

    3-1

    -0.5

    0

    0.5

    1

    -2

    0

    2

    -1

    -0.5

    0

    0.5

    1

    M

    ==

    =+=

    =+=

    ],0[

    ]2,0[],2,0[,

    )8.0,(

    sin)]}sin.9.0,3(cos[2{

    cos)]}8.0,3(cos[2{

    u

    ssexz

    uusbexy

    usbexx

    -2

    0

    2

    -2

    0

    2

    -1

    -0.5

    0

    0.5

    1

    -2

    0

    2

    -3

    -2

    -1

    0

    -2

    0

    2

    -1

    -0.5

    0

    0.5

    1

    -3

    -2

    -1

    0

    -2

    0

    2

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    62/133

    61

    H Y P E R S O N I C J E T A I R P L A N E

    M ]2,0[],0],1,0[(,

    Re

    )2/sin(.1(Re

    sin.1

    =

    +=

    =

    =

    sS

    x

    sz

    xsy

    sx

    -10

    -7.5

    -5

    -2.5

    00

    1

    2

    30

    1

    2

    3

    -10

    -7.5

    -5

    -2.5

    0

    0

    1

    2

    3

    0

    2

    4

    6

    8

    -0.5

    0

    0.5

    -0.5

    0

    0.5

    0

    2

    4

    6

    8

    -0.5

    0

    0.5

    S ( s [ 0, 0.8], = 0)

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    63/133

    62

    P L U M P V A S E

    M ]2,0[),0,3.2,0[(,.sin..cos.25.Re

    cos.cos.25.Re2

    2

    =

    =

    +=

    +=

    sS

    sz

    ssxsy

    ssxsx

    -2-1

    01

    2

    -2

    0

    2

    0

    2

    4

    6

    -2

    0

    2

    -2

    -1

    0

    1

    2

    -2

    0

    2

    0

    2

    4

    -2

    -1

    0

    1

    2

    -2

    0

    2

    - 2

    0

    2

    - 2

    0

    2

    0

    2

    4

    - 2

    0

    2

    - 2

    0

    2

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    64/133

    63

    A R R O W S

    -0.4-0.2

    00.2

    0.4

    -1

    0

    1

    2

    -0.5

    -0.25

    0

    0.25

    0.5

    -0.4-0.2

    00.2

    0.4

    -1

    0

    1

    2

    -0.4-0.20

    0.20.4

    -1

    0

    1

    2

    -0.5

    -0.25

    0

    0.25

    0.5

    -0.4-0.20

    0.20.4

    -1

    0

    1

    2

    -0.5

    -0.25

    0

    0.25

    0.5

    -0.4-0.2

    0

    0.2

    0.4

    -1

    0

    1

    2

    -0.5

    -0.25

    0

    0.25

    0.5

    -0.4-0.2

    0

    0.2

    0.4

    -1

    0

    1

    2

    -0.4

    -0.2

    0

    0.20.4

    -1

    0

    1

    2

    -0.5

    -0.25

    0

    0.25

    0.5

    -0.4

    -0.2

    0

    0.20.4

    -1

    0

    1

    2

    M ]2,0[),0),2

    ,2

    [(,

    sin.5.0

    cos)1,0,().1,0,(2

    cos.cos1).1,0,( 2

    =

    =

    =

    =

    sS

    sz

    sdelcexy

    ssexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    65/133

    64

    H Y P E R B O L I C Q U A D R A T I C C Y L I N D E R 1

    R I G H T : n = 4, m = 1 R O T A T E D n = 1, m = 0.5

    00.5

    11.5

    2

    0

    0.5

    1

    1.5

    2

    -1

    0

    1

    0

    0.5

    1

    1.5

    2

    -1

    -0.5

    0

    0.5

    1

    -1

    -0.5

    0

    0.5

    1

    -1

    0

    1

    -1

    -0.5

    0

    0.5

    1

    M ]2,0[),0],1,1[(,

    5.1

    )0,,(sin

    )0,,(cos

    =

    =

    ====

    sS

    sz

    snqy

    snqxm

    m

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    66/133

    65

    H Y P E R B O L I C Q U A D R A T I C C Y L I N D E R 2

    0

    1

    2

    0

    0 . 5

    1

    1 . 5

    2

    - 1

    0

    1

    0

    0 . 5

    1

    1 . 5

    2

    0

    0 . 5

    1

    1 . 5

    2

    0

    1

    2

    - 1

    0

    1

    0

    0 . 5

    1

    1 . 5

    2

    0

    1

    2

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    67/133

    66

    E X - C E N T R I C F U L L S P R I N G

    -1

    0

    1

    -1

    0

    1

    -1

    0

    1

    2

    3

    -1

    0

    1

    M

    ===

    ==

    ===

    ==

    ))0,1(,4

    (

    cos3.0

    )cos3.0

    ))0,1(,4

    (2.0

    )0,1(,4

    (2.0

    sSaexz

    y

    x

    sSaex

    sSaex

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    68/133

    67

    E X - C E N T R I C E M P T Y S P R I N G

    -20

    -10

    0

    10

    -10

    0

    10

    0

    20

    40

    -10

    0

    10

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    69/133

    68

    E X C E N T R I C P E N T A G O N H E L I X

    -50

    0

    50

    -50

    0

    50

    0

    50

    100

    150

    -50

    0

    50

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    70/133

    69

    C Y L I N D E R S w i t h C O L L A R S

    -20

    2

    -2

    0

    2

    -5

    0

    5

    -2

    0

    2

    -2

    0

    2

    -2

    0

    2

    -5

    0

    5

    -2

    0

    2

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    71/133

    70

    U n i c u r s a l S u p e r m a t h e m a t i c s F u n c t i o n s 1

    - 2 2 4 6 8

    - 1

    - 0 . 5

    0 . 5

    1

    ]2

    3,2

    [,

    8.4

    8.3

    5.3

    ,

    4

    5.2

    1

    ,,

    )cos.2cos

    sin.2cos.arctansin(

    cos.2cos

    sin.2cos.arctan

    ==

    ++=

    ++=

    cba

    cb

    ay

    cb

    ax

    O l d w o m a n f r o m C a r p a t i M o u n t a i n ( R o m a ni a )

    2 4 6 8 10 12

    -1

    -0.5

    0.5

    1

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    72/133

    71

    U n i c u r s a l S u p e r m a t h e m a t i c s F u n c t i o n s 2

    2 4 6 8 1 0 1 2

    -1

    - 0 . 5

    0. 5

    1

    2 4 6 8 10 12

    -1

    -0.5

    0.5

    1

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    73/133

    72

    U n i c u r s a l Su p e r m a t h e m a t i c s F u n c t i o n s 3

    1 2 3 4 5 6

    - 1

    - 0 . 5

    0 . 5

    1

    W a l k i n g P i n g u i n s

    -2.5 2.5 5 7.5 10 12.5

    -1

    -0.5

    0.5

    1

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    74/133

    73

    T o D o u b l e a n d S i m p l e C a n o e

    - 0 . 5

    - 0 . 2 5

    0

    0 . 2 5

    0 . 5

    - 1

    0

    1

    2

    - 0 . 5

    - 0 . 2 5

    0

    0 . 2 5

    0 . 5

    - 0 . 5

    - 0 . 2 5

    0

    0 . 2 5

    0 . 5

    - 1

    0

    1

    2

    - 1

    - 0 . 5

    0

    0 . 5

    1

    - 1

    0

    1

    2

    - 0 . 5

    - 0 . 2 5

    0

    0 . 2 5

    0 . 5

    - 1

    - 0 . 5

    0

    0 . 5

    1

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    75/133

    74

    R o m a n i a n f o l k d a n c e

    -4

    -2

    0

    2

    4

    -4

    -2

    0

    2

    4

    -5

    -2.5

    0

    2.5

    5

    -4

    -2

    0

    2

    4

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    76/133

    75

    P i l l o w

    2

    2 . 5

    3

    3 . 5

    4

    2

    2 . 5

    3

    3 . 5

    4

    - 1

    - 0 . 5

    0

    0 . 5

    1

    2

    2 . 5

    3

    3 . 5

    4

    2

    2 . 5

    3

    3 . 5

    4

    2

    2.5

    3

    3.5

    4

    2

    2.5

    3

    3.5

    4

    -1

    -0.5

    0

    0.5

    1

    2

    2.5

    3

    3.5

    4

    2

    2.5

    3

    3.5

    4

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    77/133

    76

    T e r r a w i t h M a r k e d M e r i d i a n s

    2

    2.5

    3

    3.5

    4

    2

    2.5

    3

    3.5

    4

    -1

    -0.5

    0

    0.5

    1

    2

    2.5

    3

    3.5

    4

    2

    2.5

    3

    3.5

    4

    M

    =

    =

    +=

    sexz

    ucexy

    cexx

    sin.

    cos.3

    , S( s u [0, 3 /2], [0, 2]

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    78/133

    77

    S u p e r m a t h e m a t i c s C o l u m n s

    ??

    x = cosq t

    y = sinq t , t [0, 2]z u [-1,1]

    -1

    -0.50

    0.51

    -1

    -0.5

    0

    0.5

    1

    -2

    0

    2

    4

    -1

    -0.5

    0

    0.5

    1

    - 0 . 5- 0 . 2 5

    00 . 2 5

    0 . 5

    - 0 . 5

    - 0 . 2 5

    0

    0 . 2 5

    0 . 5

    0

    2

    4

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    79/133

    78

    A e r o d y n a m i c S o l i d

    -

    0

    1

    - 1

    - 0 . 5

    0

    0 . 5

    1- 1

    - 0 . 5

    0

    0 . 5

    1

    - 1

    - 0 . 5

    0

    0 . 5

    1

    - 1

    0

    1

    - 1

    - 0 . 5

    0

    0 . 5

    1

    - 1

    - 0 . 5

    0

    0 .5

    1

    - 1

    - 0 . 5

    0

    0 .5

    1

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    80/133

    79

    H a l v i n g C u r v e

    x = cosq , y = sinq , with numerical ex-center : S( s, = 0 ), [0, 8 ]S (s = 1, = 0) S (s = 0.89, = 0)

    5 10 15 20 25

    -1

    -0.5

    0.5

    1

    5 10 15 20 25

    -1

    -0.5

    0.5

    1

    S (s = 0, = 0) S (s = 0.5, = 0)

    5 10 15 20 25

    -1

    -0.5

    0.5

    1

    5 10 15 20 25

    -1

    -0.5

    0.5

    1

    S (s = 0.5, = 0) , x x2 S (s = 0.5, = 0), y y 2

    5 10 15 20 25

    -1

    -0.5

    0.5

    1

    5 10 15 20 25

    -1

    -0.5

    0.5

    1

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    81/133

    80

    The crook lines (s 0) - a generalization of straight lines ( s = 0 )

    y = m. aex (x, S )=m{x-arcsin [s. sin(x - )]}, Numerical ex centre S (s, ) fixed

    m = 1, S (s [- 1, 0], = 0 ) m = 1, S (s [0, +1], = 0 )

    -3 -2 -1 1 2 3

    -3

    -2

    -1

    1

    2

    3

    -3 -2 -1 1 2 3

    -3

    -2

    -1

    1

    2

    3

    y = m. aex (x, S), Numerical ex centre S (s, ) variable: s =xx

    s

    sin.cos

    0

    - 3 - 2 - 1 1 2 3

    - 4

    - 2

    2

    4

    S ( s = s0 /cos x .sin x s0 [- 1, 0], = 0 )

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    82/133

    81

    The crook lines (s 0) - a generalization of straight lines ( s = 0 )

    y = m. aex (x, S )=m{x-arcsin [s. sin(x - )]}, Numerical ex centre S (s, ) variable

    m = 1, S (s = s0.cos2x, s0 [- 1, 0], = 0 ) m = 1, S (s= s0.cos2x, s0 [0, +1], = 0 )

    -3 -2 -1 1 2 3

    -3

    -2

    -1

    1

    2

    3

    -3 -2 -1 1 2 3

    -3

    -2

    -1

    1

    2

    3

    m = 1, S (s= s0.cos2x, s0 [-1, +1], = 0 ), x [ - 3 / 2, 3 / 2 ]

    - 4 - 2 2 4

    - 6

    - 4

    - 2

    2

    4

    6

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    83/133

    82

    The crook lines (s 0, s = 1) - a generalization of straight lines ( s = 0 )

    y = m. aex (x, S )= m{x-arcsin [s. sin(x - )]}, Numerical ex centre S (s, ) fixed

    -3 -2 -1 1 2 3

    -4

    -2

    2

    4

    Numerical ex centre S (s, ) fixed : S (s [ - 1 , 1 ] , = 1 )

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    84/133

    83

    A R A B E S Q U E S 1

    - 1 - 0 . 5 0 . 5 1

    - 1

    - 0 . 5

    0 . 5

    1

    - 1 - 0 . 5 0 . 5 1

    - 1

    - 0 . 5

    0 . 5

    1

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    85/133

    84

    S T A R S

    x = Dex 10 . Rex 10 .cos y = Dex 10 . Rex 10 .sin

    S( s [-1,1 ] , = 0), [0, 2]

    -1 -0.5 0.5 1

    -1.5

    -1

    -0.5

    0.5

    1

    1.5

    -1 -0.5 0.5 1

    -1.5

    -1

    -0.5

    0.5

    1

    1.5

    - 2 - 1 . 5 - 1 - 0 . 5 0 . 5 1 1 . 5

    - 1 . 5

    - 1

    - 0 . 5

    0 . 5

    1

    1 . 5

    x = (1-cos 5 ) cos / Rex 10 .y = (1- cos 5 ) sin /Rex 10 .

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    86/133

    85

    H y s t e r e t i c C u r v e s 1

    -4 -2 2 4

    -1

    -0.5

    0.5

    1

    x = 1

    M

    }00.1,95,0,9.0,8.0,6.0,4.0,2.0{

    ]2

    ,2

    [),0,5.0

    1],1,0[(

    ,

    cos1

    )sin())1,(,(sin

    sin1

    )cos())1,(,(cos

    22

    22

    =

    =

    =

    ===

    ===

    s

    sS

    ssSqy

    ssSqx

    yx

    y

    x

    -4 -2 2 4

    -1

    -0.5

    0.5

    1

    x =0.5

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    87/133

    86

    H y s t e r e t i c C u r v e s 2

    -4 -2 2 4

    -4

    -2

    2

    4

    x = 0.5; y=0.5=

    -4 -2 2 4

    -4

    -2

    2

    4

    x = 0.5; y= 1

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    88/133

    87

    H y s t e r e t i c C u r v e s 3

    -4 -2 2 4

    -4

    -2

    2

    4

    x = 0.5; y= 0.5

    -4 -2 2 4

    -4

    -2

    2

    4

    x = 0.2; y=0.5

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    89/133

    88

    E x C e n t r i c C i r c u l a r C u r v e s w i t h E x C e n t r i c V a r i a b l e

    - 1 - 0 . 5 0 . 5 1

    - 1

    - 0 . 5

    0 . 5

    1

    0.1 S t e p

    - 1 - 0 . 5 0 . 5 1

    - 1

    - 0 . 5

    0 . 5

    1

    0 . 2 S t e p

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    90/133

    89

    F i l i g r e e 1

    - 1 - 0 . 5 0 . 5 1

    - 1

    - 0 . 5

    0 . 5

    1

    s 0.1 u [-1, 1] with 0.1 S t e p

    - 1 - 0 . 5 0 . 5 1

    - 1

    - 0 . 5

    0 . 5

    1

    s 0.1 u [-1, 1] with 0.2 S t e p

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    91/133

    90

    F I L I G R E E 2

    - 1 - 0 . 5 0 . 5 1

    - 1

    - 0 . 5

    0 . 5

    1

    s 0.1 u [0, 1] with 0.1 S t e p s 0.1 u [-1, 0] with 0.1 S t e p

    -1 -0.5 0.5 1

    -1

    -0.5

    0.5

    1

    -1 -0.5 0.5 1

    -1

    -0.5

    0.5

    1

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    92/133

    91

    U s h a p e d C u r v e s i n 2 D a n d 3 D

    y = arctan{1/[sex . Abs[cex ]]}, S(s [-1,1] , = 0), [ 0, 2]

    1 2 3 4 5 6

    -60

    -40

    -20

    20

    40

    60

    0

    2

    4

    6

    -0.5

    0

    0.5

    -1

    0

    1

    -1

    0

    1

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    93/133

    92

    E x p l o s i o n s

    -1 -0.5 0.5 1

    -1

    -0.5

    0.5

    1

    -1 -0.5 0.5 1

    -1

    -0.5

    0.5

    1

    ]2,0[

    )0],1,0[(,

    10Re

    sin.8cos.Re

    cos.10cos.

    =

    =

    = sS

    x

    sy

    x

    sx

    ]2,0[

    )0],1,0[(,

    8Re

    sin.7cos.Re

    cos.10cos.

    =

    =

    = sS

    x

    sy

    x

    sx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    94/133

    93

    S p a t i a l F i g u r e

    M ]2,0[),0],1,0[(,

    )]cos3(,2[3

    5Re

    5

    2

    0

    =

    ==

    =

    =

    sS

    ssSCexz

    yx

    bexx

    -1

    0

    1

    -2

    0

    2

    -2

    0

    2

    -1

    0

    1

    -2

    0

    2

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    95/133

    94

    P l a n e t s a n d s t a r s

    -1 -0.5 0.5 1

    -1

    -0.5

    0.5

    1

    -2 -1 1 2

    -2

    -1

    1

    2

    -2 -1 1 2

    -2

    -1

    1

    2

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    96/133

    95

    E x C e n t r i c C i r c l e ( n = 1 ) a n d A s t e r o i d ( n = 2, 4 , 6 )

    M ]2,0[),0],1,1[(,

    =

    =

    =sS

    sexy

    cexxn

    n

    -1 -0.5 0.5 1

    -1

    -0.5

    0.5

    1

    0.2 0.4 0.6 0.8 1

    0.2

    0.4

    0.6

    0.8

    1

    n = 1 n = 2

    0.2 0.4 0.6 0.8 1

    0.2

    0.4

    0.6

    0.8

    1

    0.2 0.4 0.6 0.8 1

    0.2

    0.4

    0.6

    0.8

    1

    n = 4 n = 6

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    97/133

    96

    E x C e n t r i c A s t e r o i d ( n = 3, 5 ,7 ,9)

    ]2,0[),0],1,1[(,

    =

    =

    =sS

    sexy

    cexxn

    n

    -1 -0.5 0.5 1

    -1

    -0.5

    0.5

    1

    -1 -0.5 0.5

    -1

    -0.5

    0.5

    1

    n = 3 n = 5

    -0.4 -0.2 0.2 0.4

    -0.4

    -0.2

    0.2

    0.4

    -0.2 -0.1 0.1 0.2

    -0.2

    -0.1

    0.1

    0.2

    n = 7 n = 9

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    98/133

    97

    E x C e n t r i c L e m n i s c a t e s

    M { x = dex[, S(s = s0. cos, = - / 2]; y = dex[, S( s = s0.cos 2, = 0)] }, s0 [0, 1], [0, 2]

    0.4 0.6 0.8 1.2 1.4 1.6

    0.5

    1

    1.5

    2

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    99/133

    98

    B u t t e r f l y w i t h S y m m e t r i c a l C e n t e r 1

    M { x = dex[, S(s = s0. cos, = - / 2]; y = dex[, S( s = s0.sin 2, = 0)] }, s0 [0, 1], [0, 2]

    0.4 0.6 0.8 1.2 1.4 1.6

    0.5

    1

    1.5

    2

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    100/133

    99

    B u t t e r f l y w i t h S y m m e t r i c a l C e n t e r 2

    M { x = dex[, S(s = s0. cos, = - / 2]; y = dex[, S( s = s0.sin 2, = 0)] }, s0 [0, 1], [0, 2]

    0 . 6 0 . 8 1 . 2 1 . 4

    0 . 5

    1

    1 . 5

    2

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    101/133

    100

    B u t t e r f l y w i t h S y m m e t r i c a l C e n t e r 3

    0 . 6 0 . 8 1 . 2 1 . 4

    0 . 2 5

    0 . 5

    0 . 7 5

    1

    1 . 2 5

    1 . 5

    1 . 7 5

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    102/133

    101

    B u t t e r f l y R a p i d l y F l a p p i n g t h e W i n g s

    0.5 1 1.5 2

    0.5

    1

    1.5

    2

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    103/133

    102

    F l o w e r w i t h F o u r P e t a l e s

    0.25 0.5 0.75 1 1.25 1.5 1.75

    0.25

    0.5

    0.75

    1

    1.25

    1.5

    1.75

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    104/133

    103

    E c C e n t r i c P y r a m i d

    0.5

    1

    1.5

    0

    0.5

    1

    1.5

    2

    -1

    -0.5

    0

    0.5

    1

    0.5

    1

    1.5

    0

    0.5

    1

    1.5

    2

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    105/133

    104

    A e r o d y n a m i c P r o f i l e w i t h S u p e r m a t h e m a t i c s F u n c t i o n s 1

    M { x = 2 cex( + / 6, S( s = 0.6, = 0)), y = 2 sex(( + / 6, S( s [ -1, 0], = 0) )}, [0, 2]

    -2 -1 1 2

    -1

    -0.5

    0.5

    1

    M { x = 2 cex( + / 6, S( s = 0.6, = 0)), y = sex(( + / 6, S( s [ 0, 1], = 0) )}, [0, 2]

    -2 -1 1 2

    -1

    -0.5

    0.5

    1

    M { x = 2 cex( + / 6, S( s = 0.2, = 0)), y = sex(( + / 2 , S( s [ 0, 0.9], = 0) )}, [0, 2]

    - 1 - 0 . 5 0 . 5 1

    - 1

    - 0 . 5

    0. 5

    1

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    106/133

    105

    A e r o d y n a m i c P r o f i l e w i t h S u p e r m a t h e m a t i c s F u n c t i o n s 2

    -1 -0.5 0.5 1

    -1

    -0.5

    0.5

    1

    M { x = 2 cex( + / 6, S( s = 0.2, = 0)), y = sex(( + / , S( s [ 0, 0.9], = 0) )}, [0, 2]- 2

    - 1 . 5

    - 1

    - 0 . 5

    0- 1

    - 0 . 5

    0

    - 2

    - 1 . 5

    - 1

    - 0 . 5

    0

    - 1

    - 0 . 5

    0

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    107/133

    106

    S A L T C e l l a r

    -2

    -1

    0

    1

    2

    -2

    -1

    0

    1

    -2

    -1

    0

    1

    2

    -2

    -1

    0

    1

    2

    -2

    -1

    0

    1

    M ]2,0[),0],1,1[(,

    .2

    )2/(

    1,2

    2,1

    =+

    =

    =

    =

    sS

    sz

    rexy

    rexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    108/133

    107

    S U P E R M A T H E M A T I C S T O W E R

    -5

    -2.5

    0

    2.5

    5

    -5

    0

    5

    -4

    -2

    0

    2

    4

    -5

    -2.5

    0

    2.5

    5

    -5

    0

    5

    M ]2,0[),0],2,0[(,

    sin..2

    )).0,1(,().cos3(

    ))0,1(,().3(

    =

    ==+=

    ==+=

    sS

    sdexs

    dexsSsexsy

    sScexdexsx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    109/133

    108

    THE CONTINUOUS TRANSFORMATION OF A RIGHT TRIANGLE INTO ITS HYPOTENUSE

    -1 -0.5 0.5 1

    -1

    -0.5

    0.5

    1

    -1 -0.5 0.5 1

    -1

    -0.5

    0.5

    1

    x=cex(, S(s, = 0)), y=cex(, S(-s, =0)), S(s [0,1], = 0), [ 0, ]

    THE CONTINUOUS TRANSFORMATION OF QUADRATE (Rx = Ry)

    OR RECTANGLE (Rx Ry) INTO ITS DIAGONAL

    -1 -0.5 0.5 1

    -1

    -0.5

    0.5

    1

    x = Rx cex(, S(s, = 0)), y = Ry cex(, S(-s, =0)), S(s [ -1, 1 ], = 0), [ 0, ]

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    110/133

    109

    Mo d i f i e d c e x a n d s e x

    ]2,0[),0],1,1[(

    ),2Re

    2sin.arctansin(

    22

    =

    +==

    sS

    x

    sMsexy

    ]2,0[),0],1,1[(

    ),2Re

    2sin.arctancos(

    22

    =

    +==

    sS

    x

    sMcexy

    1 2 3 4 5 6

    -1

    -0.5

    0.5

    1

    1 2 3 4 5 6

    -1

    -0.5

    0.5

    1

    ]2,0[),0],1,1[(

    ),2Re

    2sin.arctancos(

    21

    =

    ==

    sS

    x

    sMcexy

    ]2,0[),0],1,1[(

    ),2Re

    2sin.arctansin(

    21

    =

    ==

    sS

    x

    sMsexy

    1 2 3 4 5 6

    -1

    -0.5

    0.5

    1

    1 2 3 4 5 6

    -1

    -0.5

    0.5

    1

    1 2 3 4 5 6

    -1

    -0.5

    0.5

    1

    1 2 3 4 5 6

    -1

    -0.5

    0.5

    1

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    111/133

    110

    S i n u o u s S u r f a c e w i t h A n a l i t i c a l S u p e r m a t h e m a t i c s F u n c t i o n s

    ]4,20[],2.1,4[],3

    sin5

    cos21.0arcsin[),( =

    x

    xxxf

    -4

    -2

    0

    25

    10

    15

    20

    -0.2

    -0.1

    0

    0.1

    0.2

    -4

    -2

    0

    2

    S e c t i o n f o r = 8 and x [ - 5, 10 ]

    - 1 0 1 0 2 0 3 0

    - 0 . 2

    - 0 . 1

    0 . 1

    0 . 2

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    112/133

    111

    S u p e r m a t h e m a t i c s S p i r a l

    x = 0.3 cos Exp[0.2 (0.25 arcsin(sin0.25 )y = 0.3 cos Exp[0.2 (0.25 arcsin(sin0.25 )

    -10 -5 5 10

    -10

    -5

    5

    10

    -1 -0.5 0.5 1

    -1

    -0.5

    0.5

    1

    [ 0, 80] [ 0, 40]S u p e r m a t h e m a t i c s P a r a b l e s

    2 4 6 8 10

    -3

    -2

    -1

    1

    2

    3

    ]1,1[),0],1,1[(, +=

    =

    =

    sS

    aexy

    aexx

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    113/133

    112

    A R A B E S Q U E S 2

    -6 -4 -2 2 4 6

    -1

    1

    2

    3

    4

    -6 -4 -2 2 4 6

    -1

    1

    2

    3

    4

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    114/133

    113

    S u p e r m a t h e m a t i c s f u n c t i o n s cex xy and sex xy

    cex xy = cos[xy-arcsin[s.sin (xy - ]] for s = 0.4 and s = 0.9 ;x [-3,3], y [-3,3

    1 2 3 4 5 6 7

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7

    1

    2

    3

    4

    5

    6

    7

    S (s = 0.4, = 0) S (s = 0.9, = 0)

    sex xy = sin[xy-arcsin[s.sin (xy - ]] for s = 0.4 and s = 0.9 ;x [-3,3], y [-3,3

    1 2 3 4 5 6 7

    1

    2

    3

    4

    5

    6

    7

    1 2 3 4 5 6 7

    1

    2

    3

    4

    5

    6

    7

    S (s = 0.4, = 0) S (s = 0.9, = 0)

  • 8/14/2019 Selariu SuperMathematics Functions, editor Florentin Smarandache

    115/133