hyperspace.docx

8/9/2019 HyperSpace.docx

1/21

HyperSpace, User Manual

Written by Paul Bourke

December 1990

This document is the user manual for a program on the Macintosh for exploring

dimensional geometry! "ere #e #ill introduce the concepts of D geometry$ describe

the use of the application$ and present some examples!

Introduction

%ne dimensional space consists of the points along a line$ only one number isre&uired to uni&uely describe any position in a 1D #orld!

T#o dimensional space consists of all the points on a plane$ t#o numbers are

re&uired to uni&uely describe any position in a 'D #orld! The t#o numbers used to

describe any point can be considered to be positions on t#o noncolinear one

dimensional axes$ the particular axis system from all the possible combinations isusually chosen to be the one #here the t#o axes are perpendicular to each other!

(or three dimensional space #e add another axis perpendicular to the t#o used for

the t#o dimensional space! This is #hat #e use to represent the uni)erse #e li)e in$

three numbers uni&uely describe any point in our *D uni)erse and they are usually

gi)en the symbols x$y$ and +!

,n four dimensions #e -simply- need to add another coordinate axis that is

perpendicular to the three axes used for *D space! Most people ha)e difficulty

)isuali+ing this but there is no problem mathematically and #e can approach higherdimensions in the same #ay .although #e #on/t do so here! We #ill use the symbol

# to represent the coordinate axis of this th dimension! ny point then in four

dimensional space can be represented by the four numbers #$x$y$ and +!

Hyperdimensional objects

The geometry of familiar ob2ects in higher dimensions is #ell established

mathematically but is generally outside our e)eryday experience! mathematical

approach #ill not be attempted here but rather the intuiti)e description of D

geometric ob2ects!
http://paulbourke.net/geometry/http://paulbourke.net/geometry/


2/21

Generation by intuition

The creation of D geometric ob2ects by intuition relies on being able to determine an

progressi)e description of the e&ui)alent ob2ect in *D and ho# it #as created from its

'D e&ui)alent! ,f #e -kno#- ho# the *D ob2ect is created from 'D and maybe ho#

the 'D ob2ect #as created from 1D then the rules can be applied to the *D ob2ect toform the D e&ui)alent!

,n #hat follo#s the rules for progressing each geometric primiti)e from 1 to ' and

then to * dimensions is explained! ll the geometries belo# are implemented in the

"yper3pace program$ they can be explored theer!

Cube

one dimensional cube is a line segment! To create a 'D cube$ a s&uare! #e -extrude-

the line .1D cube along the ne#ly introduced axis! To form the *D cube #e again

extrude the s&uare .'D cube along the ne#ly introduced axis! 3o the generaltechni&ue for generating a n41 dimensional cube is take the n dimensional cube and

extrude it along the ne# n41 dimension!

Tetrahedron

1D tetrahedron is a line! 'D tetrahedon is an e&uilateral triangle! The n41 dimensiontetrahedron is created by taking the midpoint of the n dimensional tetrahedron and

pulling it into the n41 dimension!


3/21

Octahedron

1D octahedron is a line! 'D octahedron is a s&uare! To form a n41 dimensional

octahedron take the midpoint of the n dimensional octahedron and pull it along the

positi)e and negati)e axis of the n41 dimension!

Pyramid

To create a n41 dimensional s&uare based pyramid take the n dimensional cube and

pull the midpoint into the n41 dimension!


4/21

Prism

n41 dimensional prism is the n dimensional cube #ith t#o opposite midpoints

pulled into the n41 dimension!

Display techniues

The main )isuali+ation problem associated #ith *D modelling is ho# to represent a

*D scene on a medium #ith one fe#er dimensions$ ie5 a computer screen or piece of

paper! This problem is made much more difficult for "yper3pace since no# #e need

to represent D ob2ects on a medium #ith t#o fe#er dimensions! "yper3pace

employs four fundamental techni&ues to sol)e this problem$ they are discussed belo#!

Parallel Projections

,n order to )isuali+e *D ob2ects on a 'D medium it is common to simply ignore one

coordinate! (or example5 ignore the + coordinate and dra# the x and y coordinates!

3uch a techni&ue gi)es #hat are called parallel pro2ections and isometric pro2ections

of *D ob2ects!

This techni&ue can be used to represent D ob2ects on a 'D medium$ ho#e)er in this

case #e ha)e to ignore t#o coordinates! (or example5 ignore the # and y coordinates

and dra# only the x and + coordinates! This is #hat the xy pro2ection menu item in the)ie# menu does! %ne should be )ery #ary about trying to )isuali+e the geometry of a

D ob2ect by this techni&ue! ,t is analogous to trying to )isuali+e the geometry of a *D

cube by pro2ections of it onto a line!

HyperCube e!amples


5/21

"le#ations and plans

nother techni&ue employed to represent *D ob2ects on a 'D medium is to pro2ect the

ob2ect onto each coordinate plane$ xy$ x+$ and y+!

This gi)es the common top6bottom .plan$ front6back .ele)ation and left6right )ie#s

of a *D ob2ect! (or example a simple *D structure is sho#n belo# #ith the

pro2ections onto the three coordinate planes!

This techni&ue can be used for D ob2ects except that there are six 'D pro2ection

planes$ namely #x$ #y$ #+$ xy$ x+$ and y+! 7ach of these can be displayed in the six

)ie#s #indo# sho#n belo#


6/21

8ote that the axes labels are placed in the topleft and bottomright of each )ie#

portion! The three )ie# portions in the top left corner correspond to the top$ front$ and

side )ie#s in *D! (or an unrotated "yper:ube the six axis )ie#s are &uite boring$

they are all s&uares! The follo#ing sho#ns the ; axis )ie#s if the "yper:ube is

rotated on all planes!


7/21

Slices

The third techni&ue used to represent *D ob2ects on a 'D medium is to contour theob2ect$ effecti)ely cutting slices along one coordinate axis! ,t is usual in *D to take the

contours along the + axis but any axis #ill do! n arbitrary slicing plane may also be

used or alternati)ely the contours can al#ays be #ith respect to one axis and the

ob2ect rotated in order to obtain arbitrary slicing planes! This process essentially

remo)es one dimension$ so slicing a *D ob2ect results in a series of 'D cur)es! the

contour lines! To contour D ob2ects #e #ill choose to al#ays slice along the # axis!


8/21

The #indo# for different numbers of slices simply partitions the #indo# into smaller

)ie# portions! The axes labels are sho#n$ the # axes is the one along #hich the slices

are taken$ the y axes is ignored .pro2ected onto the x+ plane! 8ine slices of the

"yper:ube are sho#n belo#! ,t doesn/t seem that difficult to imagine each of the

solids belo#!!!!!the problem comes #hen you need to stack them up together in the th

dimension!


9/21

Depth Cue

The last techni&ue #hich can be used in con2unction the others is to shade the edgesaccording to the )alue of one coordinate! ,n the "yper3pace application$ points #ith a

large y coordinate may be shaded red$ points #ith a small y may be shaded blue and

all points in bet#een are coloured #ith the appropriate intermediate hue!


10/21


11/21

$otatin% the object

When )ie#ing an ob2ect using a computer aided design package there are t#ooperations #hich can be useful! They are5 rotate the camera about the ob2ect or rotate

the ob2ect about its centre! There are subtle differences bet#een the t#o approaches$

the "yper3pace application implements rotation of ob2ect about any one of the

coordinate axis!

:onsidering the situation in *D$ there are three axes$ x$y$ and + about #hich an ob2ect

can be rotated! lternati)ely there can be considered to be three planes through #hich

an ob2ect may be rotated! The t#o descriptions are identical$ the only difference is in

ho# they are described! 3o$ for example$ rotation about the x axis is the same as

rotation in the y+ plane$ rotation about the y axis is rotation in the x+ plane androtation about the + axis is rotation in the xy plane! The rotation plane is the

terminology used here!


12/21

,n D there are six planes though #hich an ob2ect can be rotated!

The amount of rotation in each plane is specified #ith the rotate

controls #indo# sho#n belo#!

7ach of the six sliders ranges from 1=0 degrees at the top to 1=0

degrees at the bottom! By clicking on the arro#s the angle )ariesby one degree inter)als$ by clicking in the grey region the angle

)aries in 10 degree steps! ,f the slider reaches the bottom of thescale it #ill #rap around to the top and similarly #hen the slider

reaches the top it #ill #rap around to the bottom$ ie5 190 degree

rotation > 1=0 degree rotation! With dimension set to * the firstthree sliders are disabled!

$eadin% models &rom a &ile

,t is possible to use "yper3pace to )ie# other dimensional ob2ects! To do this a

specially formatted text file must be created describing the geometry! This file is

opened #ith the read menu item in the file menu! The format #ill be described belo#

but it should be stressed that the format must be closely adhered to if "yper3pace is to

correctly read the ob2ect geometry!

The text file consists of three sections$ they are5 the )ertex description$ the edge

description$ and finally the face description! 7ach section starts #ith the number of

elements in that section$ for example$ the )ertex section starts #ith the number of

)ertices that #ill follo#$ the edge section starts #ith the number of edges to follo#$

similarly for the faces!

?ertices5 7ach )ertex appears on a line by itself and consists of four floating point

numbers representing the #$x$y$+ coordinates!

7dges5 7ach line of the edge description contains t#o numbers representing the )ertex

numbers of the endpoints of the edge!

(aces5 7ach line of the face description consists of fi)e numbers! The first is the

number of )ertices in the face$ it may be either * or ! The remaining numbers are the)ertex numbers making up the face! ll four )ertex numbers must be specified e)en if

they are not used$ the unused )ertex number could be 0$ it #ill be ignored in any case!

The )ertices and edges must be present for any ob2ect description but the faces

information is optional! ,f the faces are not specified then the number of faces must be


13/21

set to 0! 8ote ho#e)er that #ithout faces the slicing method of )isuali+ation #ill not

#ork!

This format is the one generated #hen the sa)e menu item is used in the file menu! n

example of a correctly formatted file is sho#n belo#

5

-1 -1 -1 1

-1 1 1 1

-1 1 -1 -1

-1 -1 1 -1

1 0 0 0

10

0 1

0 2

0 3

1 2

1 3

2 3

4 0

4 1

4 2

4 3

10

3 0 1 2 0

3 0 1 3 0

3 0 2 3 03 1 2 3 0

3 4 0 1 0

3 4 1 2 0

3 4 1 3 0

3 4 0 2 0

3 4 0 3 0

This ob2ect contains @ )ertices$ 10 edges$ and 10 faces! 8ote from the edge list that the

)ertex numbers start from 0 not 1$ so the )ertex numbers in the edge and face list

range from 0 to not from 1 to @! The face list consists of only * point faces$ thefourth column must be present but it is only considered for point faces!

While all the numbers abo)e are integers floating point numbers are e&ually )alid!

The separators bet#een each number can be any -#hite- character$ most commonly

spaces or tabs #ould be used! This makes it possible to create these files #ith basic

programs$ spreadsheets$ #ord processors$ etc!


14/21

$e&erences

Experiments in 4 Dimensions$ "eiserman$ Da)id A!$ TB Books ,nc

Regular Polytypes$ :oxeter$ "!3!M!$ MacMillam :ompany

Introduction to Geometry$ :oxeter$ "!3!M!$ ohn Wiley C 3ons ,nc

4 Dimensional Space$ 7ckhart$ A!$ ,ndiana ni)ersity Press

$e%ular Polytopes 'Platonic solids(

in )DWritten byPaul Bourke

une 199E

pdate 8o)ember '00*

*uote &rom "phesians +, -./ '0in% 1ames #ersion(

That ye !!! may be able to comprehend #ith all the saints #hat is the breadth$

and length$ and depth$ and height!,s this the first reference .8e# Testament of the Bible to dimensionsF


15/21

,n) dimensionsthere are ; regular polytopes$ this is the highest number that exist in

any dimension greater than '! They are listed and described in order of increasing cell

numbers belo#! 7ach regular polytope is supplied as data in at least t#o )ersions$ the

first is asimple ascii formatlisting )ertices$ edges$ and faces! The second is as an %((

file ready for Geom?ie#!

3implex

lso kno#n aspentatopeorpentachoron

3elfdual

D e&ui)alent of the tetrahedron

@ tetrahedral cells$ 10 triangular faces$ 10 edges$ @ )ertices

* tetrahedra meet at an edge

,t can be intuiti)ely formed by choosing a point in one higher dimension

e&uidistant to all the )ertices in the current dimension and connecting this ne# point

to all the current )ertices!

The se&uence is5 Point Aine Triangle Tetrahedron 3implex

?ertices$ edges$ and faces

%(( )ersion

"ypercube lso kno#n as theTesseract

Dual #ithcross polytope

D e&ui)alent of the cube
http://paulbourke.net/geometry/hyperspace/#asciiformathttp://paulbourke.net/geometry/hyperspace/#asciiformathttp://paulbourke.net/geometry/hyperspace/5cell.asciihttp://paulbourke.net/geometry/hyperspace/5cell.offhttp://paulbourke.net/geometry/hyperspace/#crosspolytopehttp://paulbourke.net/geometry/hyperspace/#crosspolytopehttp://paulbourke.net/geometry/hyperspace/#asciiformathttp://paulbourke.net/geometry/hyperspace/5cell.asciihttp://paulbourke.net/geometry/hyperspace/5cell.offhttp://paulbourke.net/geometry/hyperspace/#crosspolytope


16/21

= cubic cells$ ' s&uare faces$ *' edges$ 1; )ertices

* cubes meet at an edge

,t can be intuiti)ely formed by mo)ing the profile in the current dimension in a

direction perpendicular to that dimension by an edge length!

The se&uence is5 Point Aine 3&uare :ube "ypercube


%(( )ersion

:ross Polytope

lso kno#n as3 cellorhe!adecachoron

Dual #ithhypercube

D e&ui)alent of the octahedron

1; tetrahedral cells$ *' triangular faces$ ' edges$ = )ertices

tetrahedra meet at an edge

The generation se&uence is5 Point Aine 3&uare %ctahedron :rosspolytope


%(( )ersion
http://paulbourke.net/geometry/hyperspace/8cell.asciihttp://paulbourke.net/geometry/hyperspace/8cell.offhttp://paulbourke.net/geometry/hyperspace/#hypercubehttp://paulbourke.net/geometry/hyperspace/#hypercubehttp://paulbourke.net/geometry/hyperspace/16cell.asciihttp://paulbourke.net/geometry/hyperspace/16cell.offhttp://paulbourke.net/geometry/hyperspace/8cell.asciihttp://paulbourke.net/geometry/hyperspace/8cell.offhttp://paulbourke.net/geometry/hyperspace/#hypercubehttp://paulbourke.net/geometry/hyperspace/16cell.asciihttp://paulbourke.net/geometry/hyperspace/16cell.off


17/21

' cell

lso kno#n as theicositetrachoron

3elfdual

8o e&ui)alent in other dimensions

' octahedral cells$ 9; triangular faces$ 9; edges$ ' )ertices

* octahedra meet at an edge


%(( )ersion

1'0 :ell

lso kno#n as thehecatonicosachoron

Dual #ith;00 cell

D e&ui)alent of the dodecahedron

1'0 dodecahedral cells$ E'0 fi)e sided faces$ 1'00 edges$ ;00 )ertices

* dodecahedra meeting per edge
http://paulbourke.net/geometry/hyperspace/24cell.asciihttp://paulbourke.net/geometry/hyperspace/24cell.offhttp://paulbourke.net/geometry/hyperspace/#600cellhttp://paulbourke.net/geometry/hyperspace/#600cellhttp://paulbourke.net/geometry/hyperspace/24cell.asciihttp://paulbourke.net/geometry/hyperspace/24cell.offhttp://paulbourke.net/geometry/hyperspace/#600cell


18/21


%(( )ersion

;00 :ell

lso kno#n as thehe!acosichoron

Dual #ith1'0 cell

D e&ui)alent to the icosahedron

;00 tetrahedral cells$ 1'00 triangular faces$ E'0 edges$ 1'0 )ertices

@ tetrahedra meeting per edge


%(( )ersion

Hi%her Dimensions

,nhi%her dimensions '4, 3, - 5555(there are only * regular polytopes in any particular

dimensionsH These * regular polytopes are the e&ui)alent of the tetrahedron$ cube$ and

octahedron in * dimensions$ they are normally called the nsimplex$ ncube$ and n

crosspolytope respecti)ely #here n stands for the dimension!

n.simple!
http://paulbourke.net/geometry/hyperspace/120cell.asciihttp://paulbourke.net/geometry/hyperspace/120cell.offhttp://paulbourke.net/geometry/hyperspace/#120cellhttp://paulbourke.net/geometry/hyperspace/#120cellhttp://paulbourke.net/geometry/hyperspace/600cell.asciihttp://paulbourke.net/geometry/hyperspace/600cell.offhttp://paulbourke.net/geometry/hyperspace/120cell.asciihttp://paulbourke.net/geometry/hyperspace/120cell.offhttp://paulbourke.net/geometry/hyperspace/#120cellhttp://paulbourke.net/geometry/hyperspace/600cell.asciihttp://paulbourke.net/geometry/hyperspace/600cell.off


19/21

lso kno#n as thehypertetrahedron

3elfdual

n 4 1 cells each of #hich is an .n1 simplex

n 4 1 )ertices

n .n 4 1 6 ' edges

n.cube

lso kno#n as thehypercube

Dual #ith ncrosspolytope

' n cells each of #hich is an .n1 cube

'n)ertices

n 'n 1edges

' n .n 1$ .n' s&uares

n.crosspolytope

lso kno#n as thehyperoctahedron

Dual #ith ncube

'ncells each of #hich is an .n1 simplex

' n )ertices

' n .n 1 edges

n 'n 1$ .n' crosspolytopes

De&initions


20/21

Dual

The Idual/ of a regular polytope is another polytope$ also regular$ ha)ing one )ertex in

the center of each cell of the polytope #e started #ith! The dual of the dual of a

regular polytope is the one #e started #ith .only smaller!

Sel&.dual,t is possible for a regular polytope to be it/s o#n dual$ for example$ all the regular

polytopes in ' dimensions! ,n * dimensions the tetrahedron is selfdual! %b)iously for

a polytope to be selfdual it must ha)e the same number of cells as )ertices!

Polyhedral &ormula

8umber of )ertices number of edges 4 number of faces number of cells > 0

6erte!78ace ascii &ormat

These files are formatted as follo#s$ hopefully the description belo# along #ith an

example from abo)e #ill gi)e all the information needed to translate the geometryinto other formats! (or more information see?7( format

The first line contains the number of )ertices

The next lines .one for each )ertex contain numbers$ each consists of the

floating point numbers being the #$x$y$+ coordinate of that )ertex! The line number

.counting from 0 is the )ertex ,D!

The next line contains the number of edges

The subse&uent lines describe each edge$ each consists of t#o )ertex ,Ds

making up that edge!

The next line contain the number of faces

The subse&uent lines describe each face! The first number of each line is the

number of )ertices in the face! The rest of the line contains a list of )ertex ,Ds making

up that face!

$e&erences


21/21

) Dimensional Space7ckhart$ A!

,ndiana ni)ersity Press

)D objects in ?7( &ormat

3ourced mostly from 3atoshi Kamaguchi and Gordon Lindlmann

:ompiled byPaul Bourke1'0cell!asci

i

1'0cell!o

ff

1'0cell1trunc!asc

ii!g+

1'0cell1trunc!of

f!g+

1'0cellsnub!ascii

!g+

1'0cellsnub!off!

g+

1;cell!ascii 1;cell!off1;cell1trunc!ascii 1;cell1trunc!off

'cell!ascii 'cell!off'cell1trunc!ascii 'cell1trunc!off'cellsnub!ascii!g+

'cellsnub!off!g+

@cell!ascii @cell!off @cell1trunc!ascii @cell1trunc!off @cellsnub!ascii @cellsnub!off

;00cell!asci

i

;00cell!o

ff

;00cell1trunc!asc

ii!g+

;00cell1trunc!of

f!g+

;00cellpluscells!

ascii

=cell!ascii =cell!off =cell1trunc!ascii =cell1trunc!off =cellsnub!ascii =cellsnub!off
http://paulbourke.net/dataformats/vef/http://paulbourke.net/geometry/http://paulbourke.net/geometry/hyperspace/120cell.asciihttp://paulbourke.net/geometry/hyperspace/120cell.asciihttp://paulbourke.net/geometry/hyperspace/120cell.offhttp://paulbourke.net/geometry/hyperspace/120cell.offhttp://paulbourke.net/geometry/hyperspace/120cell_1trunc.ascii.gzhttp://paulbourke.net/geometry/hyperspace/120cell_1trunc.ascii.gzhttp://paulbourke.net/geometry/hyperspace/120cell_1trunc.off.gzhttp://paulbourke.net/geometry/hyperspace/120cell_1trunc.off.gzhttp://paulbourke.net/geometry/hyperspace/120cell_snub.ascii.gzhttp://paulbourke.net/geometry/hyperspace/120cell_snub.ascii.gzhttp://paulbourke.net/geometry/hyperspace/120cell_snub.off.gzhttp://paulbourke.net/geometry/hyperspace/120cell_snub.off.gzhttp://paulbourke.net/geometry/hyperspace/16cell.asciihttp://paulbourke.net/geometry/hyperspace/16cell.offhttp://paulbourke.net/geometry/hyperspace/16cell_1trunc.asciihttp://paulbourke.net/geometry/hyperspace/16cell_1trunc.offhttp://paulbourke.net/geometry/hyperspace/24cell.asciihttp://paulbourke.net/geometry/hyperspace/24cell.offhttp://paulbourke.net/geometry/hyperspace/24cell_1trunc.asciihttp://paulbourke.net/geometry/hyperspace/24cell_1trunc.offhttp://paulbourke.net/geometry/hyperspace/24cell_snub.ascii.gzhttp://paulbourke.net/geometry/hyperspace/24cell_snub.ascii.gzhttp://paulbourke.net/geometry/hyperspace/24cell_snub.off.gzhttp://paulbourke.net/geometry/hyperspace/24cell_snub.off.gzhttp://paulbourke.net/geometry/hyperspace/5cell.asciihttp://paulbourke.net/geometry/hyperspace/5cell.offhttp://paulbourke.net/geometry/hyperspace/5cell_1trunc.asciihttp://paulbourke.net/geometry/hyperspace/5cell_1trunc.offhttp://paulbourke.net/geometry/hyperspace/5cell_snub.asciihttp://paulbourke.net/geometry/hyperspace/5cell_snub.offhttp://paulbourke.net/geometry/hyperspace/600cell.asciihttp://paulbourke.net/geometry/hyperspace/600cell.asciihttp://paulbourke.net/geometry/hyperspace/600cell.offhttp://paulbourke.net/geometry/hyperspace/600cell.offhttp://paulbourke.net/geometry/hyperspace/600cell_1trunc.ascii.gzhttp://paulbourke.net/geometry/hyperspace/600cell_1trunc.ascii.gzhttp://paulbourke.net/geometry/hyperspace/600cell_1trunc.off.gzhttp://paulbourke.net/geometry/hyperspace/600cell_1trunc.off.gzhttp://paulbourke.net/geometry/hyperspace/600cell_pluscells.asciihttp://paulbourke.net/geometry/hyperspace/600cell_pluscells.asciihttp://paulbourke.net/geometry/hyperspace/8cell.asciihttp://paulbourke.net/geometry/hyperspace/8cell.offhttp://paulbourke.net/geometry/hyperspace/8cell_1trunc.asciihttp://paulbourke.net/geometry/hyperspace/8cell_1trunc.offhttp://paulbourke.net/geometry/hyperspace/8cell_snub.asciihttp://paulbourke.net/geometry/hyperspace/8cell_snub.offhttp://paulbourke.net/dataformats/vef/http://paulbourke.net/geometry/http://paulbourke.net/geometry/hyperspace/120cell.asciihttp://paulbourke.net/geometry/hyperspace/120cell.asciihttp://paulbourke.net/geometry/hyperspace/120cell.offhttp://paulbourke.net/geometry/hyperspace/120cell.offhttp://paulbourke.net/geometry/hyperspace/120cell_1trunc.ascii.gzhttp://paulbourke.net/geometry/hyperspace/120cell_1trunc.ascii.gzhttp://paulbourke.net/geometry/hyperspace/120cell_1trunc.off.gzhttp://paulbourke.net/geometry/hyperspace/120cell_1trunc.off.gzhttp://paulbourke.net/geometry/hyperspace/120cell_snub.ascii.gzhttp://paulbourke.net/geometry/hyperspace/120cell_snub.ascii.gzhttp://paulbourke.net/geometry/hyperspace/120cell_snub.off.gzhttp://paulbourke.net/geometry/hyperspace/120cell_snub.off.gzhttp://paulbourke.net/geometry/hyperspace/16cell.asciihttp://paulbourke.net/geometry/hyperspace/16cell.offhttp://paulbourke.net/geometry/hyperspace/16cell_1trunc.asciihttp://paulbourke.net/geometry/hyperspace/16cell_1trunc.offhttp://paulbourke.net/geometry/hyperspace/24cell.asciihttp://paulbourke.net/geometry/hyperspace/24cell.offhttp://paulbourke.net/geometry/hyperspace/24cell_1trunc.asciihttp://paulbourke.net/geometry/hyperspace/24cell_1trunc.offhttp://paulbourke.net/geometry/hyperspace/24cell_snub.ascii.gzhttp://paulbourke.net/geometry/hyperspace/24cell_snub.ascii.gzhttp://paulbourke.net/geometry/hyperspace/24cell_snub.off.gzhttp://paulbourke.net/geometry/hyperspace/24cell_snub.off.gzhttp://paulbourke.net/geometry/hyperspace/5cell.asciihttp://paulbourke.net/geometry/hyperspace/5cell.offhttp://paulbourke.net/geometry/hyperspace/5cell_1trunc.asciihttp://paulbourke.net/geometry/hyperspace/5cell_1trunc.offhttp://paulbourke.net/geometry/hyperspace/5cell_snub.asciihttp://paulbourke.net/geometry/hyperspace/5cell_snub.offhttp://paulbourke.net/geometry/hyperspace/600cell.asciihttp://paulbourke.net/geometry/hyperspace/600cell.asciihttp://paulbourke.net/geometry/hyperspace/600cell.offhttp://paulbourke.net/geometry/hyperspace/600cell.offhttp://paulbourke.net/geometry/hyperspace/600cell_1trunc.ascii.gzhttp://paulbourke.net/geometry/hyperspace/600cell_1trunc.ascii.gzhttp://paulbourke.net/geometry/hyperspace/600cell_1trunc.off.gzhttp://paulbourke.net/geometry/hyperspace/600cell_1trunc.off.gzhttp://paulbourke.net/geometry/hyperspace/600cell_pluscells.asciihttp://paulbourke.net/geometry/hyperspace/600cell_pluscells.asciihttp://paulbourke.net/geometry/hyperspace/8cell.asciihttp://paulbourke.net/geometry/hyperspace/8cell.offhttp://paulbourke.net/geometry/hyperspace/8cell_1trunc.asciihttp://paulbourke.net/geometry/hyperspace/8cell_1trunc.offhttp://paulbourke.net/geometry/hyperspace/8cell_snub.asciihttp://paulbourke.net/geometry/hyperspace/8cell_snub.off

hyperspace.docx

Documents