Polygons and polyhedra over finite field.math.nsc.ru/conference/g2/g2c2/LavshukT.pdf · Regular...
Transcript of Polygons and polyhedra over finite field.math.nsc.ru/conference/g2/g2c2/LavshukT.pdf · Regular...
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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
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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
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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
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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
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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
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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
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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
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Square over F5
Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 8 / 27
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Regular pentagon over F19
Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 9 / 27
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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
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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
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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
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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
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Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 13 / 27
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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
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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
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Regular tetrahedron in F 347
Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 15 / 27
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Regular tetrahedron in F 347
Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 16 / 27
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Regular tetrahedron in F 373
Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 17 / 27
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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
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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
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Regular tetrahedron in F 37
Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 19 / 27
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Regular tetrahedron in F 317
Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 20 / 27
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Regular tetrahedron in F 341
Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 21 / 27
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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
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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
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Regular dodecahedron in F 319
Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 23 / 27
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Regular dodecahedron in F 359
Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 24 / 27
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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
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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
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Regular icosahedron in F 329
Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 26 / 27
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Regular icosahedron in F 371
Tamara Lavshuk (Novosibirsk State University Sobolev Institute of Mathemathics)Polygons and polyhedra over �nite �eld. September, 2014 27 / 27