Euclidean geometry is incidence geometry

8th ANZMC

John Bamberg

Centre for the Mathematics of Symmetry and Computation,The University of Western Australia

December 2014

Incidence Geometry

Objects (points, lines, planes, etc)

Symmetric reflexive incidence relation between them.

Two incident objects of the same type are equal.

Projective plane

Objects are points and lines.

Axiom 1. Two points are incident with a unique line.

Axiom 2. Two lines are incident with a unique point.

Axiom 3. There exist four points, no three collinear.

Hyperbolic planes

Menger (1938,1940,1946,1971), Abbott (1941,1942,1944),DeBaggis (1946,1948), Jenks (1940).

1 Two points are incident with a unique line.

2 Each line is on at least one point.

3 There exist three non-collinear points.

4 If the point P is on neither of two intersecting lines ` and m, then P is on a line that meets ` but not m.

5 Of three collinear points, at least one has the property that every line through it intersects at least one ofeach pair of intersecting lines through the other two.

6 If P is not on `, then there exist two distinct lines on P not meeting ` and such that each line meeting `meets at least one of those two lines.

7 Any two non-collinear rays have a common parallel line.

8 Pascal’s Theorem on rimpoints.

9 Pappus’ Theorem.

10 Desargues Theorem.

Theorem (Skala 1992)Every model of the above axioms is a hyperbolic plane and conversely.

Thus hyperbolic geometry is incidence geometry.

Hyperbolic planes

Menger (1938,1940,1946,1971), Abbott (1941,1942,1944),DeBaggis (1946,1948), Jenks (1940).

1 Two points are incident with a unique line.

2 Each line is on at least one point.

3 There exist three non-collinear points.

4 If the point P is on neither of two intersecting lines ` and m, then P is on a line that meets ` but not m.

5 Of three collinear points, at least one has the property that every line through it intersects at least one ofeach pair of intersecting lines through the other two.

6 If P is not on `, then there exist two distinct lines on P not meeting ` and such that each line meeting `meets at least one of those two lines.

7 Any two non-collinear rays have a common parallel line.

8 Pascal’s Theorem on rimpoints.

9 Pappus’ Theorem.

10 Desargues Theorem.

Theorem (Skala 1992)Every model of the above axioms is a hyperbolic plane and conversely.

Thus hyperbolic geometry is incidence geometry.

Hyperbolic planes

Menger (1938,1940,1946,1971), Abbott (1941,1942,1944),DeBaggis (1946,1948), Jenks (1940).

1 Two points are incident with a unique line.

2 Each line is on at least one point.

3 There exist three non-collinear points.

4 If the point P is on neither of two intersecting lines ` and m, then P is on a line that meets ` but not m.

5 Of three collinear points, at least one has the property that every line through it intersects at least one ofeach pair of intersecting lines through the other two.

6 If P is not on `, then there exist two distinct lines on P not meeting ` and such that each line meeting `meets at least one of those two lines.

7 Any two non-collinear rays have a common parallel line.

8 Pascal’s Theorem on rimpoints.

9 Pappus’ Theorem.

10 Desargues Theorem.

Theorem (Skala 1992)Every model of the above axioms is a hyperbolic plane and conversely.

Thus hyperbolic geometry is incidence geometry.

“Menger (1938) has shown that in hyperbolic geometry theconcepts of betweenness and equidistance can be defined in termsof the single notion of point-line incidence, and thus that planehyperbolic geometry can be axiomatized in terms of this notionalone by rephrasing a traditional axiom system in terms ofincidence alone. This was one of the most important discoveries,for it shed light on the true nature of hyperbolic geometry, whichmoved nearer to projective geometry than to its one-time sisterEuclidean geometry, in partial opposition to which it had been born.Menger even claimed that this fact alone proved – pace Poincare –that hyperbolic geometry was actually simpler than Euclideangeometry.”

– Victor Pambuccian, 2006.

DesarguesianPappian

TopologicalPappian

Pythagorean

Euclidean

Tarski

Real

OrderedDesarguesian

Hilbert planes

Undefined concepts: point, line, incidence, between and congruent.

Incidence axioms

I-1. Every pair of points is incident with a unique line.

I-2. Every line is incident with at least two points.

I-3. There exist three points no 3 collinear.

Betweenness axioms

B-1. If B is between A and C , then A,B,C are distinct collinearpoints, and B is between C and A.

B-2. Given two points B and D, there exist points A, C and E suchthat B is between A and C , C is between B and D, and D isbetween B and E .

B-3. A,B,C collinear =⇒ one is between the other two.

B-4. Given three non-collinear points A,B, and C and a line ` noton any of the three points: being on the same side of ` istransitive and if A and B are not on the same side of ` and B,and C are not on the same side of `, then A and B are on thesame side of `.

Congruence axioms

C-1. If A and B are points, and A′ is any point, then for each ray remanating from A′ there is a unique point B ′ on r such thatAB is congruent to A′B ′.

C-2. Congruence of line segments is transitive and reflexive.

C-3. If B is between A and C , B ′ is between A′ and B ′, AB iscongruent to A′B ′ and BC is congruent to B ′C ′, then AC iscongruent to A′C ′.

C-4. Given an angle ∠BAC and a ray A′B ′ emanating from A′,

there is a unique ray−−→A′C ′ on a given side of the line A′B ′ such

that ∠BAC is congruent to ∠B ′A′C ′.

C-5. Congruence of angles is transitive and reflexive.

C-6. Side-angle-side implies congruence for triangles.

A Hilbert planeis a model for our incidence, betweenness and congruence axioms.

Parallel axiomFor every line ` and every point P not incident with `, there is atmost one line m incident with p which is parallel to `.

`

Pythagorean planeHilbert plane + parallel axiom

A Hilbert planeis a model for our incidence, betweenness and congruence axioms.

Parallel axiomFor every line ` and every point P not incident with `, there is atmost one line m incident with p which is parallel to `.

`

Pythagorean planeHilbert plane + parallel axiom

A Hilbert planeis a model for our incidence, betweenness and congruence axioms.

Parallel axiomFor every line ` and every point P not incident with `, there is atmost one line m incident with p which is parallel to `.

`

Pythagorean planeHilbert plane + parallel axiom

Dedekind’s axiomSuppose that the set of all points on a line ` is the disjoint unionof two nonempty subsets such that no point of either subset isbetween two points of the other. Then there exist a unique pointO on ` such that one of the subsets is a ray emanating from O.

Theorem (Borsuk and Szmielew 1960)

A Pythagorean plane is isomorphic to R2 if and only if it satisfiesDedekind’s axiom.

Dedekind’s axiomSuppose that the set of all points on a line ` is the disjoint unionof two nonempty subsets such that no point of either subset isbetween two points of the other. Then there exist a unique pointO on ` such that one of the subsets is a ray emanating from O.

Theorem (Borsuk and Szmielew 1960)

A Pythagorean plane is isomorphic to R2 if and only if it satisfiesDedekind’s axiom.

Circle with centre O and radius OASet of all points P with OP congruent to OA.

Inside and outside a circleLet K be a circle with centre O and radius OA. A point Q is . . .

inside if OQ < OA;

outside if OA < OQ.

Circle-circle intersection propertyGiven two circles, if one of them is on both a point inside and apoint outside the other, then they will meet.

Euclidean planePythagorean + Circle-circle intersection property

Circle with centre O and radius OASet of all points P with OP congruent to OA.

Inside and outside a circleLet K be a circle with centre O and radius OA. A point Q is . . .

inside if OQ < OA;

outside if OA < OQ.

Circle-circle intersection propertyGiven two circles, if one of them is on both a point inside and apoint outside the other, then they will meet.

Euclidean planePythagorean + Circle-circle intersection property

Division ringField

Topological field

Pythagorean

Euclidean

Real-closed

Real

Ordered

R R ∩ A Constructiblenumbers

Hilbertfield

Q

C

Zp H

Hilbert’stwistedLaurentseries

A field is . . .

formally real if −1 is not a sum of squares in the field.

real closed if it is formally real but no proper extension of itis formally real.

Euclidean if it is formally real and every element is either asquare or the negative of a square.

Theorem (Tarski 1959)

A Hilbert plane is Euclidean ⇐⇒ it is isomorphic to a Cartesianplane over a Euclidean field.

CorollaryTwo Euclidean planes are isomorphic ⇐⇒ correspondingEuclidean fields are isomorphic.

A field is . . .

formally real if −1 is not a sum of squares in the field.

real closed if it is formally real but no proper extension of itis formally real.

Euclidean if it is formally real and every element is either asquare or the negative of a square.

Theorem (Tarski 1959)

A Hilbert plane is Euclidean ⇐⇒ it is isomorphic to a Cartesianplane over a Euclidean field.

CorollaryTwo Euclidean planes are isomorphic ⇐⇒ correspondingEuclidean fields are isomorphic.

Similarity of R2

Preserves distance up to a scalar:

d(xg , yg ) = λ · d(x , y).

Theorem (Caratheodory 1937)

A bijection g : R2 → R2

takes circles to circles ⇐⇒ g is a similarity.

Incidence geometry of point and lines of Euclidean plane → affine plane

What if you study the incidence geometry of points and circles in aEuclidean plane?

What is a line?Set of points of the form

{P,Q} ∪ {R : {P,Q,R} are not concircular} .

Parallel lines: equal or disjoint;

Tangent circles: ∃ unique point incident with both;

Line tangent to circle: ∃ unique point incident with both;

Line secant to circle: exactly two points incident with both.

Incidence geometry of point and lines of Euclidean plane → affine plane

What if you study the incidence geometry of points and circles in aEuclidean plane?

What is a line?Set of points of the form

{P,Q} ∪ {R : {P,Q,R} are not concircular} .

Parallel lines: equal or disjoint;

Tangent circles: ∃ unique point incident with both;

Line tangent to circle: ∃ unique point incident with both;

Line secant to circle: exactly two points incident with both.

Incidence geometry of point and lines of Euclidean plane → affine plane

What if you study the incidence geometry of points and circles in aEuclidean plane?

What is a line?Set of points of the form

{P,Q} ∪ {R : {P,Q,R} are not concircular} .

Parallel lines: equal or disjoint;

Tangent circles: ∃ unique point incident with both;

Line tangent to circle: ∃ unique point incident with both;

Line secant to circle: exactly two points incident with both.

Whole lotta axioms

1 Any two distinct points are on exactly one line.2 Parallel axiom.3 There exist 3 non-collinear points.4 Every circle is incident with at least 3 points.5 Every 3 non-collinear points are incident with a unique circle.6 For every circle K and points P,Q with P incident with K

and Q not incident with K , either Q is incident with a linetangent to K at P or Q is incident with a circle tangent to Kat P (but not both).

PKQ

7 For every line ` and points P,Q with P incident with ` and Qnot incident with `, Q is on a unique circle tangent to ` at P.

8 There is a circle K such that every point is on 0, 1 or 2tangent lines to K , points on two tangent lines to K exist,and every such point lies only on secant lines to K .

9 Miquel’s Theorem. Given points A, B, C , D, E , F , G , Hsuch that {A,B,C ,D}, {A,B,E ,F}, {B,C ,F ,G},{C ,D,G ,H}, {A,D,E ,H} are concircular, we have{E ,F ,G ,H} concircular.

B

F

GH

E

C

D

A

Betweenness Q is between P and R if P,Q,R are collinearand there is a circle K incident with P and R with Qon no tangent line to K .

Congruence by orbits of Aut(π)

two point pairs are congruent if there exists anelement of Aut(π) taking the first to the second;two pairs of intersecting lines are congruent ifthere exists an element of Aut(π) taking thefirst to the second.

TheoremEvery model of axioms 1 through 9 is a Euclidean plane andconversely every Euclidean plane is a model of axioms 1 through 9.

Proof

van der Waerden-Smid (1935):Axioms 1 – 8 =⇒ affine plane AG(2,F ),

every circle is a non-degenerate conic ofAG(2,F ), disjoint from the line at infinity.

Since there are points on two tangent lines to K , we have char F 6= 2.

Since internal points exist, F has a quadratic extension E .

There exists a non-square in F .

Let n be an arbitrary non-square in F . We work in the projectivecompletion PG(2,F ). Any two non-degenerate conics are equivalent, sowe may assume that K is

x2 − ny 2 − z2 = 0.

The line z = 0 is external =⇒ its pole (0, 0, 1) is internal.

Since x = 0 is on this internal point, it is a secant line, so −n a square.

Since n was arbitrary, F is Euclidean, for this also implies −1 6= �.

Definitions of betweenness and congruence match with those in theEuclidean plane F 2, so we have a Euclidean plane, isomorphic to F 2.

The converse is straightforward, noting that the last part of Axiom 9 isjust the Line-Circle axiom applied to K .

Proof

van der Waerden-Smid (1935):Axioms 1 – 8 =⇒ affine plane AG(2,F ),

every circle is a non-degenerate conic ofAG(2,F ), disjoint from the line at infinity.

Since there are points on two tangent lines to K , we have char F 6= 2.

Since internal points exist, F has a quadratic extension E .

There exists a non-square in F .

Let n be an arbitrary non-square in F . We work in the projectivecompletion PG(2,F ). Any two non-degenerate conics are equivalent, sowe may assume that K is

x2 − ny 2 − z2 = 0.

The line z = 0 is external =⇒ its pole (0, 0, 1) is internal.

Since x = 0 is on this internal point, it is a secant line, so −n a square.

Since n was arbitrary, F is Euclidean, for this also implies −1 6= �.

Definitions of betweenness and congruence match with those in theEuclidean plane F 2, so we have a Euclidean plane, isomorphic to F 2.

The converse is straightforward, noting that the last part of Axiom 9 isjust the Line-Circle axiom applied to K .

Proof

van der Waerden-Smid (1935):Axioms 1 – 8 =⇒ affine plane AG(2,F ),

every circle is a non-degenerate conic ofAG(2,F ), disjoint from the line at infinity.

Since there are points on two tangent lines to K , we have char F 6= 2.

Since internal points exist, F has a quadratic extension E .

There exists a non-square in F .

Let n be an arbitrary non-square in F . We work in the projectivecompletion PG(2,F ). Any two non-degenerate conics are equivalent, sowe may assume that K is

x2 − ny 2 − z2 = 0.

The line z = 0 is external =⇒ its pole (0, 0, 1) is internal.

Since x = 0 is on this internal point, it is a secant line, so −n a square.

Since n was arbitrary, F is Euclidean, for this also implies −1 6= �.

Definitions of betweenness and congruence match with those in theEuclidean plane F 2, so we have a Euclidean plane, isomorphic to F 2.

The converse is straightforward, noting that the last part of Axiom 9 isjust the Line-Circle axiom applied to K .

Proof

van der Waerden-Smid (1935):Axioms 1 – 8 =⇒ affine plane AG(2,F ),

every circle is a non-degenerate conic ofAG(2,F ), disjoint from the line at infinity.

Since there are points on two tangent lines to K , we have char F 6= 2.

Since internal points exist, F has a quadratic extension E .

There exists a non-square in F .

Let n be an arbitrary non-square in F . We work in the projectivecompletion PG(2,F ). Any two non-degenerate conics are equivalent, sowe may assume that K is

x2 − ny 2 − z2 = 0.

The line z = 0 is external =⇒ its pole (0, 0, 1) is internal.

Since x = 0 is on this internal point, it is a secant line, so −n a square.

Since n was arbitrary, F is Euclidean, for this also implies −1 6= �.

Definitions of betweenness and congruence match with those in theEuclidean plane F 2, so we have a Euclidean plane, isomorphic to F 2.

The converse is straightforward, noting that the last part of Axiom 9 isjust the Line-Circle axiom applied to K .

Proof

van der Waerden-Smid (1935):Axioms 1 – 8 =⇒ affine plane AG(2,F ),

every circle is a non-degenerate conic ofAG(2,F ), disjoint from the line at infinity.

Since there are points on two tangent lines to K , we have char F 6= 2.

Since internal points exist, F has a quadratic extension E .

There exists a non-square in F .

Let n be an arbitrary non-square in F . We work in the projectivecompletion PG(2,F ). Any two non-degenerate conics are equivalent, sowe may assume that K is

x2 − ny 2 − z2 = 0.

The line z = 0 is external =⇒ its pole (0, 0, 1) is internal.

Since x = 0 is on this internal point, it is a secant line, so −n a square.

Since n was arbitrary, F is Euclidean, for this also implies −1 6= �.

Definitions of betweenness and congruence match with those in theEuclidean plane F 2, so we have a Euclidean plane, isomorphic to F 2.

The converse is straightforward, noting that the last part of Axiom 9 isjust the Line-Circle axiom applied to K .

Proof

van der Waerden-Smid (1935):Axioms 1 – 8 =⇒ affine plane AG(2,F ),

every circle is a non-degenerate conic ofAG(2,F ), disjoint from the line at infinity.

Since there are points on two tangent lines to K , we have char F 6= 2.

Since internal points exist, F has a quadratic extension E .

There exists a non-square in F .

Let n be an arbitrary non-square in F . We work in the projectivecompletion PG(2,F ). Any two non-degenerate conics are equivalent, sowe may assume that K is

x2 − ny 2 − z2 = 0.

The line z = 0 is external =⇒ its pole (0, 0, 1) is internal.

Since x = 0 is on this internal point, it is a secant line, so −n a square.

Since n was arbitrary, F is Euclidean, for this also implies −1 6= �.

Definitions of betweenness and congruence match with those in theEuclidean plane F 2, so we have a Euclidean plane, isomorphic to F 2.

The converse is straightforward, noting that the last part of Axiom 9 isjust the Line-Circle axiom applied to K .

Proof

van der Waerden-Smid (1935):Axioms 1 – 8 =⇒ affine plane AG(2,F ),

every circle is a non-degenerate conic ofAG(2,F ), disjoint from the line at infinity.

Since there are points on two tangent lines to K , we have char F 6= 2.

Since internal points exist, F has a quadratic extension E .

There exists a non-square in F .

Let n be an arbitrary non-square in F . We work in the projectivecompletion PG(2,F ). Any two non-degenerate conics are equivalent, sowe may assume that K is

x2 − ny 2 − z2 = 0.

The line z = 0 is external =⇒ its pole (0, 0, 1) is internal.

Since x = 0 is on this internal point, it is a secant line, so −n a square.

Since n was arbitrary, F is Euclidean, for this also implies −1 6= �.

Definitions of betweenness and congruence match with those in theEuclidean plane F 2, so we have a Euclidean plane, isomorphic to F 2.

The converse is straightforward, noting that the last part of Axiom 9 isjust the Line-Circle axiom applied to K .

Thus Euclidean planes are (point-circle) incidence planes.

CorollaryThe models of axioms 1 through 9 are precisely the Cartesianplanes over Euclidean fields.

Corollary (Generalised Caratheodory theorem)

Let F be a Euclidean field, and φ : F 2 → F 2 be a bijection. Thenφ takes circles to circles if and only if φ is a similarity ofd((x1, x2), (y1, y2)) = (x1 − y1)2 + (x2 − y2)2.

CorollaryThe only model of axioms 1 through 9 and Dedekind’s axiom is R2.

Thus Euclidean geometry is incidence geometry.

Thus Euclidean planes are (point-circle) incidence planes.

CorollaryThe models of axioms 1 through 9 are precisely the Cartesianplanes over Euclidean fields.

Corollary (Generalised Caratheodory theorem)

Let F be a Euclidean field, and φ : F 2 → F 2 be a bijection. Thenφ takes circles to circles if and only if φ is a similarity ofd((x1, x2), (y1, y2)) = (x1 − y1)2 + (x2 − y2)2.

CorollaryThe only model of axioms 1 through 9 and Dedekind’s axiom is R2.

Thus Euclidean geometry is incidence geometry.

Thus Euclidean planes are (point-circle) incidence planes.

CorollaryThe models of axioms 1 through 9 are precisely the Cartesianplanes over Euclidean fields.

Corollary (Generalised Caratheodory theorem)

Let F be a Euclidean field, and φ : F 2 → F 2 be a bijection. Thenφ takes circles to circles if and only if φ is a similarity ofd((x1, x2), (y1, y2)) = (x1 − y1)2 + (x2 − y2)2.

CorollaryThe only model of axioms 1 through 9 and Dedekind’s axiom is R2.

Thus Euclidean geometry is incidence geometry.