Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular...
Transcript of Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular...
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular Polytopes
Laura Mancinska
University of Waterloo,Department of C&O
January 23, 2008
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Outline
How many regular polytopes are there in n dimensions?
Definitions and examples
Platonic solids
Why only five?How to describe them?
Regular polytopes in 4 dimensions
Regular polytopes in higher dimensions
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Outline
How many regular polytopes are there in n dimensions?
Definitions and examples
Platonic solids
Why only five?How to describe them?
Regular polytopes in 4 dimensions
Regular polytopes in higher dimensions
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Polytope is the general term of the sequence“point, segment, polygon, polyhedron,. . . ”
Definition
A polytope in Rn is a finite, convex region enclosed by a finitenumber of hyperplanes. We denote it by Πn.
Examples n = 0, 1, 2, 3, 4.
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Polytope is the general term of the sequence“point, segment, polygon, polyhedron,. . . ”
Definition
A polytope in Rn is a finite, convex region enclosed by a finitenumber of hyperplanes. We denote it by Πn.
Examples n = 0, 1, 2, 3, 4.
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Polytope is the general term of the sequence“point, segment, polygon, polyhedron,. . . ”
Definition
A polytope in Rn is a finite, convex region enclosed by a finitenumber of hyperplanes. We denote it by Πn.
Examples n = 0, 1, 2, 3, 4.
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Definition
Regular polytope is a polytope Πn (n ≥ 3) with
1 regular facets
2 regular vertex figures
We define all Π0 and Π1 to be regular. The regularity of Π2 isunderstood in the usual sense.
Vertex figure at vertex v is a Πn−1 obtained by joining themidpoints of adjacent edges incident to v.
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Definition
Regular polytope is a polytope Πn (n ≥ 3) with
1 regular facets
2 regular vertex figures
We define all Π0 and Π1 to be regular. The regularity of Π2 isunderstood in the usual sense.
Vertex figure at vertex v is a Πn−1 obtained by joining themidpoints of adjacent edges incident to v.
![Page 9: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/9.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Definition
Regular polytope is a polytope Πn (n ≥ 3) with
1 regular facets
2 regular vertex figures
We define all Π0 and Π1 to be regular. The regularity of Π2 isunderstood in the usual sense.
Vertex figure at vertex v is a Πn−1 obtained by joining themidpoints of adjacent edges incident to v.
![Page 10: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/10.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Definition
Regular polytope is a polytope Πn (n ≥ 3) with
1 regular facets
2 regular vertex figures
We define all Π0 and Π1 to be regular. The regularity of Π2 isunderstood in the usual sense.
Vertex figure at vertex v is a Πn−1 obtained by joining themidpoints of adjacent edges incident to v.
![Page 11: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/11.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Star-polygons
9 52 = 9 7
2 = 9 73 =
9 83 = 9 9
2 = 9 94 =
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Kepler-Poinsot solids
95, 52 = 93, 5
2 =
9 52 , 5= 9 5
2 , 3=
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Two dimensional case
In 2 dimensions there is an infinite number of regular polytopes(polygons).
83< 84< 85< 86<
87< 88< 89< 810<
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Necessary condition in 3D
Polyhedron {p, q}Faces of polyhedron are polygons {p}Vertex figures are polygons {q}. Note that this means thatexactly q faces meet at each vertex.
(π − 2π
p
)q < 2π
1− 2p
<2q
12
<1p
+1q
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Necessary condition in 3D
Polyhedron {p, q}Faces of polyhedron are polygons {p}Vertex figures are polygons {q}. Note that this means thatexactly q faces meet at each vertex.
(π − 2π
p
)q < 2π
1− 2p
<2q
12
<1p
+1q
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Necessary condition in 3D
Polyhedron {p, q}Faces of polyhedron are polygons {p}Vertex figures are polygons {q}. Note that this means thatexactly q faces meet at each vertex.
(π − 2π
p
)q < 2π
1− 2p
<2q
12
<1p
+1q
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Necessary condition in 3D
Polyhedron {p, q}Faces of polyhedron are polygons {p}Vertex figures are polygons {q}. Note that this means thatexactly q faces meet at each vertex.
(π − 2π
p
)q < 2π
1− 2p
<2q
12
<1p
+1q
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Solutions of the inequality
Inequality
Faces are polygons {p}Exactly q faces meet at each vertex
12
<1p
+1q
Solutions
p = 3 p = 4 p = 5q = 3, 4, 5 q = 3 q = 3
But do the corresponding polyhedrons really exist?
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Solutions of the inequality
Inequality
Faces are polygons {p}Exactly q faces meet at each vertex
12
<1p
+1q
Solutions
p = 3 p = 4 p = 5
q = 3, 4, 5 q = 3 q = 3
But do the corresponding polyhedrons really exist?
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Solutions of the inequality
Inequality
Faces are polygons {p}Exactly q faces meet at each vertex
12
<1p
+1q
Solutions
p = 3 p = 4 p = 5q = 3, 4, 5
q = 3 q = 3
But do the corresponding polyhedrons really exist?
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Solutions of the inequality
Inequality
Faces are polygons {p}Exactly q faces meet at each vertex
12
<1p
+1q
Solutions
p = 3 p = 4 p = 5q = 3, 4, 5 q = 3 q = 3
But do the corresponding polyhedrons really exist?
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Solutions of the inequality
Inequality
Faces are polygons {p}Exactly q faces meet at each vertex
12
<1p
+1q
Solutions
p = 3 p = 4 p = 5q = 3, 4, 5 q = 3 q = 3
But do the corresponding polyhedrons really exist?
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
{p, q} = {4, 3}
(±1,±1,±1)
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Cube
{p, q} = {4, 3}
(±1,±1,±1)
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
{p, q} = {3, 4}
(±1, 0, 0)(0,±1, 0)(0, 0,±1)
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Octahedron
{p, q} = {3, 4}
(±1, 0, 0)(0,±1, 0)(0, 0,±1)
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
{p, q} = {3, 3}
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Tetrahedron
{p, q} = {3, 3}
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Tetrahedron
{p, q} = {3, 3}
(+1,+1,+1)(+1,−1,−1)(−1,+1,−1)(−1,−1,+1)
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
{p, q} = {3, 5}
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Icosahedron
{p, q} = {3, 5}
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Icosahedron
{p, q} = {3, 5}
(0,±τ,±1)(±1, 0,±τ)(±τ,±1, 0)
where
τ =1 +
√5
2
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
{p, q} = {5, 3}
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Dodecahedron
{p, q} = {5, 3}
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Dodecahedron
{p, q} = {5, 3}
(±1,±1,±1)(0,±τ,± 1
τ )(± 1
τ , 0,±τ)(±τ,± 1
τ , 0)where
τ =1 +
√5
2
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Five Platonic solids
Cube
84, 3<Tetrahedron
83, 3<Icosahedron
83, 5<
Octahedron
83, 4<Dodecahedron
85, 3<
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Schlafli symbol
86< 83, 4<
Desired properties of a Schlafli symbol of a regular polytope Πn
1 Schlafli symbol is an ordered set of n− 1 natural numbers2 If Πn has Schlafli symbol {k1, k2 . . . , kn−1}, then its
Facets have Schlafli symbol {k1, k2 . . . , kn−2}.Vertex figures have Schlafli symbol {k2, k3 . . . , kn−1}.
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Schlafli symbol
86< 83, 4<
Desired properties of a Schlafli symbol of a regular polytope Πn
1 Schlafli symbol is an ordered set of n− 1 natural numbers
2 If Πn has Schlafli symbol {k1, k2 . . . , kn−1}, then its
Facets have Schlafli symbol {k1, k2 . . . , kn−2}.Vertex figures have Schlafli symbol {k2, k3 . . . , kn−1}.
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Schlafli symbol
86< 83, 4<
Desired properties of a Schlafli symbol of a regular polytope Πn
1 Schlafli symbol is an ordered set of n− 1 natural numbers2 If Πn has Schlafli symbol {k1, k2 . . . , kn−1}, then its
Facets have Schlafli symbol {k1, k2 . . . , kn−2}.Vertex figures have Schlafli symbol {k2, k3 . . . , kn−1}.
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Schlafli symbol
Claim
Vertex figure of a facet is a facet of a vertex figure.
If Π4 is a regular polytope, then it has
3-dimensional facets {p, q}3-dimensional vertex figures {v, r}
We define the Schlafli symbol of Π4 to be {p, q, r}.
In general if Πn is a regular polytope, then it has
facets {k1, k2, . . . , kn−2}vertex figures {k2, . . . , kn−2, kn−1}
Thus the Schlafli symbol of Πn is {k1, k2, . . . , kn−1}.
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Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Schlafli symbol
Claim
Vertex figure of a facet is a facet of a vertex figure.
If Π4 is a regular polytope, then it has
3-dimensional facets {p, q}3-dimensional vertex figures {v, r}
We define the Schlafli symbol of Π4 to be {p, q, r}.
In general if Πn is a regular polytope, then it has
facets {k1, k2, . . . , kn−2}vertex figures {k2, . . . , kn−2, kn−1}
Thus the Schlafli symbol of Πn is {k1, k2, . . . , kn−1}.
![Page 42: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/42.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Schlafli symbol
Claim
Vertex figure of a facet is a facet of a vertex figure.
If Π4 is a regular polytope, then it has
3-dimensional facets {p, q}3-dimensional vertex figures {v, r}
We define the Schlafli symbol of Π4 to be {p, q, r}.
In general if Πn is a regular polytope, then it has
facets {k1, k2, . . . , kn−2}vertex figures {k2, . . . , kn−2, kn−1}
Thus the Schlafli symbol of Πn is {k1, k2, . . . , kn−1}.
![Page 43: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/43.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Schlafli symbol
Claim
Vertex figure of a facet is a facet of a vertex figure.
If Π4 is a regular polytope, then it has
3-dimensional facets {p, q}3-dimensional vertex figures {q, r}
We define the Schlafli symbol of Π4 to be {p, q, r}.
In general if Πn is a regular polytope, then it has
facets {k1, k2, . . . , kn−2}vertex figures {k2, . . . , kn−2, kn−1}
Thus the Schlafli symbol of Πn is {k1, k2, . . . , kn−1}.
![Page 44: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/44.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Schlafli symbol
Claim
Vertex figure of a facet is a facet of a vertex figure.
If Π4 is a regular polytope, then it has
3-dimensional facets {p, q}3-dimensional vertex figures {q, r}
We define the Schlafli symbol of Π4 to be {p, q, r}.
In general if Πn is a regular polytope, then it has
facets {k1, k2, . . . , kn−2}vertex figures {k2, . . . , kn−2, kn−1}
Thus the Schlafli symbol of Πn is {k1, k2, . . . , kn−1}.
![Page 45: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/45.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Schlafli symbol
Claim
Vertex figure of a facet is a facet of a vertex figure.
If Π4 is a regular polytope, then it has
3-dimensional facets {p, q}3-dimensional vertex figures {q, r}
We define the Schlafli symbol of Π4 to be {p, q, r}.
In general if Πn is a regular polytope, then it has
facets {k1, k2, . . . , kn−2}vertex figures {k2, . . . , kn−2, kn−1}
Thus the Schlafli symbol of Πn is {k1, k2, . . . , kn−1}.
![Page 46: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/46.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Schlafli symbol
Claim
Vertex figure of a facet is a facet of a vertex figure.
If Π4 is a regular polytope, then it has
3-dimensional facets {p, q}3-dimensional vertex figures {q, r}
We define the Schlafli symbol of Π4 to be {p, q, r}.
In general if Πn is a regular polytope, then it has
facets {k1, k2, . . . , kn−2}vertex figures {k2, . . . , kn−2, kn−1}
Thus the Schlafli symbol of Πn is {k1, k2, . . . , kn−1}.
![Page 47: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/47.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Schlafli symbol
Claim
Vertex figure of a facet is a facet of a vertex figure.
If Π4 is a regular polytope, then it has
3-dimensional facets {p, q}3-dimensional vertex figures {q, r}
We define the Schlafli symbol of Π4 to be {p, q, r}.
In general if Πn is a regular polytope, then it has
facets {k1, k2, . . . , kn−2}vertex figures {k2, . . . , kn−2, kn−1}
Thus the Schlafli symbol of Πn is {k1, k2, . . . , kn−1}.
![Page 48: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/48.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular 4-dimensional polytopes
Regular polyhedrons
{3, 3}, {3, 4}, {3, 5}, {4, 3}, {5, 3}
By superimposing we can form the following Schlafli symbols:
{3, 3, 3}, {3, 3, 4}, {3, 3, 5}{3, 4, 3}{3, 5, 3}
{4, 3, 3}, {4, 3, 4}, {4, 3, 5}{5, 3, 3}, {5, 3, 4}, {5, 3, 5}
![Page 49: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/49.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular 4-dimensional polytopes
Regular polyhedrons
{3, 3}, {3, 4}, {3, 5}, {4, 3}, {5, 3}
By superimposing we can form the following Schlafli symbols:
{3, 3, 3}, {3, 3, 4}, {3, 3, 5}
{3, 4, 3}{3, 5, 3}
{4, 3, 3}, {4, 3, 4}, {4, 3, 5}{5, 3, 3}, {5, 3, 4}, {5, 3, 5}
![Page 50: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/50.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular 4-dimensional polytopes
Regular polyhedrons
{3, 3}, {3, 4}, {3, 5}, {4, 3}, {5, 3}
By superimposing we can form the following Schlafli symbols:
{3, 3, 3}, {3, 3, 4}, {3, 3, 5}{3, 4, 3}
{3, 5, 3}{4, 3, 3}, {4, 3, 4}, {4, 3, 5}{5, 3, 3}, {5, 3, 4}, {5, 3, 5}
![Page 51: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/51.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular 4-dimensional polytopes
Regular polyhedrons
{3, 3}, {3, 4}, {3, 5}, {4, 3}, {5, 3}
By superimposing we can form the following Schlafli symbols:
{3, 3, 3}, {3, 3, 4}, {3, 3, 5}{3, 4, 3}{3, 5, 3}
{4, 3, 3}, {4, 3, 4}, {4, 3, 5}{5, 3, 3}, {5, 3, 4}, {5, 3, 5}
![Page 52: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/52.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular 4-dimensional polytopes
Regular polyhedrons
{3, 3}, {3, 4}, {3, 5}, {4, 3}, {5, 3}
By superimposing we can form the following Schlafli symbols:
{3, 3, 3}, {3, 3, 4}, {3, 3, 5}{3, 4, 3}{3, 5, 3}
{4, 3, 3}, {4, 3, 4}, {4, 3, 5}
{5, 3, 3}, {5, 3, 4}, {5, 3, 5}
![Page 53: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/53.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular 4-dimensional polytopes
Regular polyhedrons
{3, 3}, {3, 4}, {3, 5}, {4, 3}, {5, 3}
By superimposing we can form the following Schlafli symbols:
{3, 3, 3}, {3, 3, 4}, {3, 3, 5}{3, 4, 3}{3, 5, 3}
{4, 3, 3}, {4, 3, 4}, {4, 3, 5}{5, 3, 3}, {5, 3, 4}, {5, 3, 5}
![Page 54: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/54.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular 4-dimensional polytopes
Regular polyhedrons
{3, 3}, {3, 4}, {3, 5}, {4, 3}, {5, 3}
By superimposing we can form the following Schlafli symbols:
{3, 3, 3}, {3, 3, 4}, {3, 3, 5}{3, 4, 3}{3, 5, 3}
{4, 3, 3}, {4, 3, 4}, {4, 3, 5}{5, 3, 3}, {5, 3, 4}, {5, 3, 5}
![Page 55: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/55.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular 4-dimensional polytopes
Regular polyhedrons
{3, 3}, {3, 4}, {3, 5}, {4, 3}, {5, 3}
By superimposing we can form the following Schlafli symbols:
{3, 3, 3}, {3, 3, 4}, {3, 3, 5}{3, 4, 3}{3, 5, 3}
{4, 3, 3}, {4, 3, 4}, {4, 3, 5}{5, 3, 3}, {5, 3, 4}, {5, 3, 5}
![Page 56: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/56.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular 5-dimensional polytopes
Six regular 4-dimensional polytopes
{3, 3, 3}, {3, 3, 4}, {3, 3, 5}, {3, 4, 3}, {4, 3, 3}, {5, 3, 3}
By superimposing we can form the following Schlafli symbols:
{3, 3, 3, 3}, {3, 3, 3, 4}, {3, 3, 3, 5}{3, 3, 4, 3}{3, 4, 3, 3}
{4, 3, 3, 3}, {4, 3, 3, 4}, {4, 3, 3, 5}{5, 3, 3, 3}, {5, 3, 3, 4}, {5, 3, 3, 5}
![Page 57: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/57.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular 5-dimensional polytopes
Six regular 4-dimensional polytopes
{3, 3, 3}, {3, 3, 4}, {3, 3, 5}, {3, 4, 3}, {4, 3, 3}, {5, 3, 3}
By superimposing we can form the following Schlafli symbols:
{3, 3, 3, 3}, {3, 3, 3, 4}, {3, 3, 3, 5}
{3, 3, 4, 3}{3, 4, 3, 3}
{4, 3, 3, 3}, {4, 3, 3, 4}, {4, 3, 3, 5}{5, 3, 3, 3}, {5, 3, 3, 4}, {5, 3, 3, 5}
![Page 58: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/58.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular 5-dimensional polytopes
Six regular 4-dimensional polytopes
{3, 3, 3}, {3, 3, 4}, {3, 3, 5}, {3, 4, 3}, {4, 3, 3}, {5, 3, 3}
By superimposing we can form the following Schlafli symbols:
{3, 3, 3, 3}, {3, 3, 3, 4}, {3, 3, 3, 5}{3, 3, 4, 3}
{3, 4, 3, 3}{4, 3, 3, 3}, {4, 3, 3, 4}, {4, 3, 3, 5}{5, 3, 3, 3}, {5, 3, 3, 4}, {5, 3, 3, 5}
![Page 59: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/59.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular 5-dimensional polytopes
Six regular 4-dimensional polytopes
{3, 3, 3}, {3, 3, 4}, {3, 3, 5}, {3, 4, 3}, {4, 3, 3}, {5, 3, 3}
By superimposing we can form the following Schlafli symbols:
{3, 3, 3, 3}, {3, 3, 3, 4}, {3, 3, 3, 5}{3, 3, 4, 3}{3, 4, 3, 3}
{4, 3, 3, 3}, {4, 3, 3, 4}, {4, 3, 3, 5}{5, 3, 3, 3}, {5, 3, 3, 4}, {5, 3, 3, 5}
![Page 60: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/60.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular 5-dimensional polytopes
Six regular 4-dimensional polytopes
{3, 3, 3}, {3, 3, 4}, {3, 3, 5}, {3, 4, 3}, {4, 3, 3}, {5, 3, 3}
By superimposing we can form the following Schlafli symbols:
{3, 3, 3, 3}, {3, 3, 3, 4}, {3, 3, 3, 5}{3, 3, 4, 3}{3, 4, 3, 3}
{4, 3, 3, 3}, {4, 3, 3, 4}, {4, 3, 3, 5}
{5, 3, 3, 3}, {5, 3, 3, 4}, {5, 3, 3, 5}
![Page 61: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/61.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular 5-dimensional polytopes
Six regular 4-dimensional polytopes
{3, 3, 3}, {3, 3, 4}, {3, 3, 5}, {3, 4, 3}, {4, 3, 3}, {5, 3, 3}
By superimposing we can form the following Schlafli symbols:
{3, 3, 3, 3}, {3, 3, 3, 4}, {3, 3, 3, 5}{3, 3, 4, 3}{3, 4, 3, 3}
{4, 3, 3, 3}, {4, 3, 3, 4}, {4, 3, 3, 5}{5, 3, 3, 3}, {5, 3, 3, 4}, {5, 3, 3, 5}
![Page 62: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/62.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular 5-dimensional polytopes
Six regular 4-dimensional polytopes
{3, 3, 3}, {3, 3, 4}, {3, 3, 5}, {3, 4, 3}, {4, 3, 3}, {5, 3, 3}
By superimposing we can form the following Schlafli symbols:
{3, 3, 3, 3}, {3, 3, 3, 4}, {3, 3, 3, 5}{3, 3, 4, 3}{3, 4, 3, 3}
{4, 3, 3, 3}, {4, 3, 3, 4}, {4, 3, 3, 5}{5, 3, 3, 3}, {5, 3, 3, 4}, {5, 3, 3, 5}
![Page 63: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/63.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Regular 5-dimensional polytopes
Six regular 4-dimensional polytopes
{3, 3, 3}, {3, 3, 4}, {3, 3, 5}, {3, 4, 3}, {4, 3, 3}, {5, 3, 3}
By superimposing we can form the following Schlafli symbols:
{3, 3, 3, 3}, {3, 3, 3, 4}, {3, 3, 3, 5}{3, 3, 4, 3}{3, 4, 3, 3}
{4, 3, 3, 3}, {4, 3, 3, 4}, {4, 3, 3, 5}{5, 3, 3, 3}, {5, 3, 3, 4}, {5, 3, 3, 5}
![Page 64: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/64.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Three regular 5-dimensional polytopes
{3, 3, 3, 3}, {3, 3, 3, 4}, {4, 3, 3, 3}
Proceeding in the same manner we can form the following Schlaflisymbols:
αn = {3, 3, . . . , 3, 3} = {3n−1} Simplex
βn = {3, 3, . . . , 3, 4} = {3n−2, 4} Cross polytope
γn = {4, 3, . . . , 3, 3} = {4, 3n−2} Hypercube
We can also get {4, 3, . . . , 3, 4} = {4, 3n−3, 4}, but it turns out tobe a honeycomb.
![Page 65: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/65.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Three regular 5-dimensional polytopes
{3, 3, 3, 3}, {3, 3, 3, 4}, {4, 3, 3, 3}
Proceeding in the same manner we can form the following Schlaflisymbols:
αn = {3, 3, . . . , 3, 3} = {3n−1} Simplex
βn = {3, 3, . . . , 3, 4} = {3n−2, 4} Cross polytope
γn = {4, 3, . . . , 3, 3} = {4, 3n−2} Hypercube
We can also get {4, 3, . . . , 3, 4} = {4, 3n−3, 4}, but it turns out tobe a honeycomb.
![Page 66: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/66.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Three regular 5-dimensional polytopes
{3, 3, 3, 3}, {3, 3, 3, 4}, {4, 3, 3, 3}
Proceeding in the same manner we can form the following Schlaflisymbols:
αn = {3, 3, . . . , 3, 3} = {3n−1} Simplex
βn = {3, 3, . . . , 3, 4} = {3n−2, 4} Cross polytope
γn = {4, 3, . . . , 3, 3} = {4, 3n−2} Hypercube
We can also get {4, 3, . . . , 3, 4} = {4, 3n−3, 4}, but it turns out tobe a honeycomb.
![Page 67: Laura Mancinska University of Waterloo, Department of C&Ohome.lu.lv/~sd20008/papers/essays/Regular Polytopes... · Regular polytope is a polytope Π n (n ≥ 3) with 1 regular facets](https://reader033.fdocuments.in/reader033/viewer/2022042922/5f6c0f721e02ee537b079ba8/html5/thumbnails/67.jpg)
Introduction Two dimensions Three dimensions Schlafli symbol Four dimensions Five and more dimensions
Summary
Dimension 1 2 3 4 ≥ 5Number of polytopes 1 ∞ 5 6 3