All faces all edges all corners are the same They are composed of regular 2D polygons There were infinitely many 2D n gons How many of these regular 3D solids are there Making a Corner for a Platonic ID: 688964
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Slide1
The (regular, 3D) Platonic Solids
All faces, all edges, all corners, are the same.They are composed of regular 2D polygons:
There were infinitely many 2D n-
gons
!
How many of these regular 3D solids are there?Slide2
Making a Corner for a Platonic
SolidPut at least 3 polygons around a shared vertexto form a real physical 3D corner!
Putting 3 squares around a vertexleaves a large (90º) gap;Forcefully closing this gap
makes the structure pop out into 3D space,forming the corner of a cube.
We can also do this with 3 pentagons: dodecahedron.Slide3
Lets try to build all possible ones:
from triangles: 3, 4, or 5 around a corner:from squares: only 3 around a corner:
from pentagons: only 3 around a corner:from hexagons:
“floor tiling”, does not
bend!higher
n-gons: do not fit
around a vertex without undulations (forming saddles);
Now
the edges are no longer all alike!
Why
Only 5 Platonic Solids?Slide4
Utah Teapot
The “Sixth Platonic Solid”-- if you belong to the Computer Graphics community.Slide5
Constructing a 4D Corner:
creates a 3D corner creates a 4D corner
?
2D
3D
4D
3D
Forcing closure:Slide6
How Do We Find All 4D Polytopes?
Reasoning by analogy helps a lot:-- How did we find all the Platonic solids?
Now: Use the Platonic solids as “tiles” and ask:What can we build from tetrahedra?or from cubes?or from the other 3 Platonic solids?
Need to look at dihedral angles
:Tetrahedron: 70.5°, Octahedron: 109.5
°, Cube: 90°,
Dodecahedron: 116.5°, Icosahedron: 138.2°.Slide7
All Regular Polytopes in 4D
Using Tetrahedra (70.5°): 3
around an edge (211.5°) (5 cells)
Simplex 4 around an edge (282.0
°) (16 cells) Cross-Polytope
5 around an edge (352.5°
) (600 cells) 600-Cell
Using Cubes (90
°
):
3
around an edge (270.0
°
)
(8 cells) Hypercube Using Octahedra (109.5°):
3 around an edge (328.5°) (24 cells)
24-Cell Using Dodecahedra (116.5°):
3 around an edge (349.5°) (120 cells
) 120-CellUsing Icosahedra (138.2°):
None! : dihedral angle is too large
( 414.6°).Slide8
Wire-Frame Projections
Project 4D polytope from 4D space to 3D space:Shadow of a solid object is mostly a “blob”.
Better to use wire frame, so we can also see what is going on
at the back side. Slide9
Oblique or Perspective Projections
3D Cube 2D 4D Cube 3D
( 2D )
We may use color to give “depth” information.Slide10
Constructing 4D Regular Polytopes
Let's construct all 4D regular polytopes-- or rather, “good” projections of them. What is a “good” projection ?
Maintain as much of the symmetry as possible;Get a good feel for the structure of the polytope.What are our options ? A parade of various projections
Slide11
5-Cell or
“4D Simplex”
5 cells (tetrahedra),10 faces (triangles),
10 edges, 5 vertices.(Perspective projection)Slide12
16-Cell or
“4D-Cross Polytope”16 cells (tetrahedra), 32 faces,
24 edges, 8 vertices.Slide13
4D-Hypercube or
“Tessaract” 8 cells (cubes), 24 faces (squares), 32 edges,
16 vertices.Slide14
24-Cell
24 cells(octahedra), 96 faces, 96
edges, 24 vertices.1152 symmetries!Slide15
24-Cell
Showing four congruent Hamiltonian cycles (in red, green, yellow, blue),each visiting every vertex exactly once.Slide16
120-Cell
120 cells (dodecahedra), 720 faces (pentagons), 1200 edges,
600 vertices.Slide17
600-Cell
600 cells (tetrahedra),1200 faces,
720 edges, 120 vertices.Slide18
Polytopes in Higher Dimensions
Use 4D tiles, look at “dihedral” angles between cells:5-Cell: 75.5°
, Tessaract: 90°, 16-Cell: 120
°, 24-Cell: 120°
, 120-Cell: 144°, 600-Cell: 164.5°.
Most 4D polytopes are too round …But we can use 3 or 4
5-Cells, and 3
Tessaracts
.
There
are
three methods
by
which we can generate regular polytopes for 5D and
for all higher dimensions.Slide19
“Dihedral Angles in Higher Dim.”
Consider the angle through which one cell has to be rotated to be brought in coincidence with
an adjoining neighbor cell.
Space
2D
3D
4D
5D
6D
Simplex
Series
60
°
70.5
°
75.5
°
78.5
°
80.4
°
90
°
Cross Polytopes
90
°
109.5
°
120
°
126.9
°
131.8
°
180
°
Measure
Polytopes
90
°
90
°
90
°
90
°
90
°
90
°Slide20
Beyond 4 Dimensions …
What happens in higher dimensions ?How many regular polytopes are therein 5, 6, 7, … dimensions ?
Only THREE
for each dimension!Pictures for 6D space:Simplex with D+1 vertices,
Hypercube with 2D vertices,Cross-Polytope with 2D vertices.Slide21
Hypercube Series
“Measure Polytope” Series
Consecutive
perpendicular sweeps:
1D 2D 3D 4D
This series
extends
to arbitrary dimensions!Slide22
4D Hypercube
Vertex-first ProjectionSlide23
Preferred Hypercube Projections
Use Cavalier Projections to maintain sense of parallel sweeps:Slide24
6D Hypercube
Oblique ProjectionSlide25
6D Zonohedron
Sweep symmetrically in 6 directions (in 3D)Slide26
Simplex Series
Connect all the dots among n+1 equally spaced vertices:(Find next one above COG).
1D 2D 3D
This series also goes on indefinitely!
The issue is how to make “nice” projections.Slide27
5D Simplex: 6 Vertices
Two methods:
Avoid central intersection:
Offset edges from middle.
Based on Tetrahedron
(plus 2 vertices inside).
Based on OctahedronSlide28
6D Simplex: 7
VerticesStart from 5D arrangement that avoids central edge intersection.
Then add point in center:
= skewed octahedron with center vertexSlide29
7D and 8D
SimplicesUse a warped cube to avoid intersecting diagonalsSlide30
Cross Polytope
Series
Place vertices on all
coordinate half-axes
,
a unit-distance away from origin.
Connect all vertex pairs that lie on different axes.
1D
2D
3D
4D
A square frame for
every pair of axes
6 square frames
= 24 edges
(=Duals of Measure Polytopes)Slide31
4D Cross Polytope
… another model with three interwoven Hamiltonian cycles.Slide32
5D Cross
Polytope (10 vertices)
Warped cube + 2 verticesSlide33
6D Cross Polytope
12 vertices
i
cosahedral symmetry
(but edge-intersections)Slide34
7D Cross Polytope
14 vertices cube + octahedronSlide35
5D and Beyond
Dim.
#
The three polytopes that result from theSimplex series,Cross polytope series,Measure polytope series,. . . is all there is in 5D and beyond!
2D 3D 4D
5D 6D 7D 8D 9D … 5 6
3 3 3 3 3 3Luckily, we live in one of the interesting dimensions!
Duals !