Polygons and polyhedra over �nite �eld.

Tamara Lavshuk

Novosibirsk State University

Sobolev Institute of Mathemathics

September, 2014

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 1 / 27

Regular stars of order n (Wildberger N. J.)

De�nition:

A regular star of order n is a list of n distinct mutually non-parallel lines

[l0, l1, . . . , ln−1] such that lk+1 = Σlk (lk−1) for all k , with the convention

that lk ≡ lk+n for all k , and the convention that

[l0, l1, . . . ln−2, ln−1] ≡ [l1, l2, . . . ln−1, l0][l0, l1, . . . ln−2, ln−1] ≡ [ln−1, ln−2 . . . l1, l0]

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 2 / 27

Regular n-gons

Suppose that [l0, l1, . . . , ln−1] is a regular star of order n with common

intersection O. Choose a point A0 on l0 distinct from O. De�ne the

sequence

A0, A2 = σl1(A0), A4 = σl3(A2), A6 = σl5(A4), . . .

and so on, so that lies on l2k by induction.

Connect the points by the lines sequentially.

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 3 / 27

Regular polygons over �nite �eld Fp

De�nition:

If list of lines [l0, l1, . . . , ln−1] is a regular star of order n with the common

intersection O of the lines, one obtains by this construction a set of npoints(vertices)with the convention that the squared distances between its

(adjacent) vertices are all equal modulo p, and lines which pass through

these points (sides) is called a regular polygon over �nite �eld Fp.

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 4 / 27

The following results are obtained by Norman Wildberger:

Theorem 2.1 (Order three star)A regular star of order three exists precisely when the number 3 is a

non-zero square.

Theorem 2.2 (Order �ve star)A regular star of order �ve exists precisely when there is a non-zero number

r satisfying the conditions

i) r2 = 5,ii) 2(5− r)

is the square.

Theorem 2.3 (Order seven star)A regular star of order �ve exists precisely when there is a non-zero number

r for which

7− 56s + 112s2 − 64s3 = 0

and such that s(1− s) is squareTamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 5 / 27

Regular triangle over �nite �eld Fp

Let ∆ABC be a regular triangle over R, with the coordinates of the vertices

A(−12 ;√32 ), B(−1

2 ;−√32 ), C (1; 0)

In the �eld F11:

A(5; 3), B(5; 8), C (1; 0)

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 6 / 27

Regular triangle over �nite �eld Fp

Let ∆ABC be a regular triangle over R, with the coordinates of the vertices

A(−12 ;√32 ), B(−1

2 ;−√32 ), C (1; 0)

In the �eld F11:

A(5; 3), B(5; 8), C (1; 0)

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 7 / 27

Square over F5

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 8 / 27

Regular pentagon over F19

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 9 / 27

Regular hexagon over F11 and regular octagon over F7

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 10 / 27

Irregular triangle with the coordinates

A1(6; 2), A2(1; 0), A3(6, 11)

|A1A2|2 = 29, |A2A3|2 = 146, |A3A1|2 = 81

In the �eld F13:

|A1A2|2 ≡ |A2A3|2 ≡ |A3A1|2 ≡ 3 (mod 13)

∆A1A2A3 is regular triangle over �nite �eld F13

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 11 / 27

Regular polyhedra over �nite �eld Fp

REGULAR HEXAHEDRON OVER FINITE FIELD Fp

De�nition:

A regular hexahedron in F 3p is a set of points and lines which form the 6

squares over Fp, any two of which intersects along a common side and two

common vertices or disjoint . Every vertex of the such squares is a vertex of

the other two.

Theorem 1:

A regular hexahedron exists over any �eld Fp

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 12 / 27

Regular polyhedra over �nite �eld Fp

REGULAR HEXAHEDRON OVER FINITE FIELD Fp

De�nition:

A regular hexahedron in F 3p is a set of points and lines which form the 6

squares over Fp, any two of which intersects along a common side and two

common vertices or disjoint . Every vertex of the such squares is a vertex of

the other two.

Theorem 1:

A regular hexahedron exists over any �eld Fp

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 12 / 27

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 13 / 27

Regular polyhedra over �nite �eld Fp

REGULAR TETRAHEDRON OVER FINITE FIELD Fp

De�nition:

A regular tetrahedron in F 3p is a set of four regular triangles over Fp

intersecting pairwise along a common side.

Theorem 2:

A regular tetrahedron exists over any �eld Fp

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 14 / 27

Regular polyhedra over �nite �eld Fp

REGULAR TETRAHEDRON OVER FINITE FIELD Fp

De�nition:

A regular tetrahedron in F 3p is a set of four regular triangles over Fp

intersecting pairwise along a common side.

Theorem 2:

A regular tetrahedron exists over any �eld Fp

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 14 / 27

Regular tetrahedron in F 347

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 15 / 27

Regular tetrahedron in F 347

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 16 / 27

Regular tetrahedron in F 373

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 17 / 27

Regular polyhedra over �nite �eld Fp

REGULAR OCTAHEDRON OVER FINITE FIELD Fp

De�nition:

A regular octahedron in F 3p is a set of points and lines which form the 8

regular triangles over Fp, any two of which intersects along a common side

and two common vertices or disjoint . Every vertex of the such triangles is

a vertex of the other three.

Theorem 3:

A regular octahedron exists over any �eld Fp

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 18 / 27

Regular polyhedra over �nite �eld Fp

REGULAR OCTAHEDRON OVER FINITE FIELD Fp

De�nition:

A regular octahedron in F 3p is a set of points and lines which form the 8

regular triangles over Fp, any two of which intersects along a common side

and two common vertices or disjoint . Every vertex of the such triangles is

a vertex of the other three.

Theorem 3:

A regular octahedron exists over any �eld Fp

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 18 / 27

Regular tetrahedron in F 37

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 19 / 27

Regular tetrahedron in F 317

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 20 / 27

Regular tetrahedron in F 341

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 21 / 27

Regular polyhedra over �nite �eld Fp

REGULAR DODECAHEDRON OVER FINITE FIELD Fp

De�nition:

A regular dodecahedron in F 3p is a set of points and lines which form the 12

regular pentagons over Fp, any two of which intersects along a common

side and two common vertices or disjoint . Every vertex of the such

pentagons is a vertex of the other two.

Theorem 4:

A regular dodecahedron exists over the �nite �eld Fp for p ≡ {0, 1, 4}(mod 5)

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 22 / 27

Regular polyhedra over �nite �eld Fp

REGULAR DODECAHEDRON OVER FINITE FIELD Fp

De�nition:

A regular dodecahedron in F 3p is a set of points and lines which form the 12

regular pentagons over Fp, any two of which intersects along a common

side and two common vertices or disjoint . Every vertex of the such

pentagons is a vertex of the other two.

Theorem 4:

A regular dodecahedron exists over the �nite �eld Fp for p ≡ {0, 1, 4}(mod 5)

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 22 / 27

Regular dodecahedron in F 319

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 23 / 27

Regular dodecahedron in F 359

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 24 / 27

Regular polyhedra over �nite �eld Fp

REGULAR ICOSAHEDRON OVER FINITE FIELD Fp

De�nition:

A regular icosahedron in F 3p is a set of points and lines which form the 20

regular triangles over Fp, any two of which intersects along a common side

and two common vertices or has a one common vertex or disjoint . Every

vertex of the such triangles is a vertex of the other four.

Theorem 5:

A regular icosahedron exists over the �nite �eld Fp for p ≡ {0, 1, 4}(mod 5)

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 25 / 27

Regular polyhedra over �nite �eld Fp

REGULAR ICOSAHEDRON OVER FINITE FIELD Fp

De�nition:

A regular icosahedron in F 3p is a set of points and lines which form the 20

regular triangles over Fp, any two of which intersects along a common side

and two common vertices or has a one common vertex or disjoint . Every

vertex of the such triangles is a vertex of the other four.

Theorem 5:

A regular icosahedron exists over the �nite �eld Fp for p ≡ {0, 1, 4}(mod 5)

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 25 / 27

Regular icosahedron in F 329

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 26 / 27

Regular icosahedron in F 371

Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 27 / 27