Added Dimensions
Visualizing Higher Dimensions
Course Outline
Lecture 1 – 4/8/2015: Visualizing Higher Dimensions
- Flatland
Lecture 2 – 4/15/2015: Einstein’s 4-D Universe
Lecture 3 – 4/22/2015: Efforts to Unify Gravity and
Quantum Physics: String
Theory and M-Theory in 10
and 11 Dimensions
Lecture 4 – 4/29/2015: Reserved – TBA
Bibliography
Abbott, Edwin. Flatland: A Romance of Many Dimensions. Millenium, 2015
(originally published 1884)
Einstein, Albert. The Meaning of Relativity. Princeton U., 1955
Yau, Shing-Tung. The Shape of Inner Space: String Theory and the Geometry
of the Universe’s Hidden Dimensions. Basic Books, 2010
Greene, Brian. The Elegant Universe: Superstrings, Hidden Dimensions and
the Quest for the Ultimate Theory. W. W. Norton,1999
Greene, Brian. The Fabric of the Cosmos. Random House/Vintage, 2004.
Greene, Brian. The Hidden Reality. Knopf, 2011
Steinhardt, Paul and Neil Turok. Endless Universe: Beyond the Big Bang.
Doubleday, 2007
Randall, Lisa. Warped Passages: Unraveling the Mysteries of the Universe’s
Hidden Dimensions. HarperCollins/Ecco, 2005
Visualizing higher dimensions
Methods of visualizing higher dimensional
objects in lower dimensional spaces
Visualizing higher dimensional objects is basically a problem
of reducing the dimensionality of the information by methods
like the following:
- Perspective projection
- Unfolding
- Use of false color
- Stereographic projection
- Taking cross-sections
Perspective projection
- Known to ancient Greeks (Anaxagoras, Democritus)
… skenographia (skenographia) – used in theater sets
- Known to Arabs (Alhazen, 1027); rediscovered by Giotto
- Angles and lengths become distorted
- Parallel lines converge to one or more vanishing points
Ascending
and
Descending,
M. C. Escher
Waterfall,
M. C. Escher
Relativity, M. C. Escher
Dragon,
M. C. Escher
2-D perspective projection of 3-D cube
and unfolded cube
(2-D projection of) 3-D perspective
projection of 4-D hypercube
“Unfolded” 4-D hypercube
Christus Hypercubus
Salvador Dali
How the hypercube unfolds
Animated 3-D perspective projection
of 4-D hypercube
3-D perspective projection of rotating
4-D hypercube
Animation of 3-D slice through
4-D hypercube
-This slice grows along the “vertical” edges connecting the
“farther” (central) cube to the “nearer” (outside) cube
Animation of 3-D slice through
4-D hypercube
-This slice grows from the “farthest” edge to the “closest”
edge, so we see a triangular prism growing. Notice how the
top and bottom of the prism sweep out the top and bottom
cubes of the hypercube
Uses of color: Knots in 4-D space
- You can untie a 3-D knot in 4-D space without cutting it
- In this sequence, darker green represents a direction
in 4-D that is farther away from our 3-D space
- The green and white parts of the knot occupy different
parts of the 4-D space and so do not intersect each other
Uses of color: Klein bottle
- The “inside” of the bottle is connected to the outside
through the 4th dimension without the bottle passing
through itself (i.e. the bottle has no “inside”)
- Darker green represents a direction in 4-D that is farther
away from our position in 3-D space
Visualizing a 3-D “hypercircle” from
within 2 dimensions
time
Visualizing a 3-D “hypercircle” from
within 2 dimensions
time
Visualizing a 3-D “hypercircle” from
within 2 dimensions
time
Visualizing a 3-D “hypercircle” from
within 2 dimensions
time
Visualizing a 3-D “hypercircle” from
within 2 dimensions
time
Visualizing a 4-D hypersphere from
within 3 dimensions
time
Visualizing a 4-D hypersphere from
within 3 dimensions
time
Visualizing a 4-D hypersphere from
within 3 dimensions
time
Visualizing a 4-D hypersphere from
within 3 dimensions
time
Visualizing a 4-D hypersphere from
within 3 dimensions
time
Visualizing a 4-D hypersphere from
within 3 dimensions
time
Visualizing a 4-D hypersphere from
within 3 dimensions
time
Visualizing a 4-D hypersphere from
within 3 dimensions
time
Visualizing a 4-D hypersphere from
within 3 dimensions
time
3-D projection of a 4-D hypersphere
3-D projection of a 4-D hypersphere
Stereographic projection
S
- Known to ancient Greeks (Hipparchus, Ptolemy)
- N projects to a point at infinity
- S projects to itself (contacts with plane)
- Great circles through N project to straight lines
- Other circles project to circles
Stereographic projection of great circles
N
-Notice that as the circle moves through the north pole (N)
the projection goes to infinity (becomes a straight line)
Stereographic projection map of the world
Using stereographic projection
to visualize higher dimensional objects
-To visualize a complex 3-D object in 2-D, first project the
planes and vertices of the object onto a 3-D sphere, then
project that projection onto the 2-D plane using stereographic
projection
3-D cube projected onto 3-D sphere
-Demo of stereographic projection:
http://torus.math.uiuc.edu/jms/java/stereop/
3-D dodecahedron
- 12 pentagonal faces
3-D dodecahedron projected onto
3-D sphere
Stereographic projection of 3-D
dodecahedron onto 2-D plane
Using stereographic projection
to visualize higher dimensional objects
-To visualize a complex 3-D object in 2-D, first project the
planes and vertices of the object onto a 3-D sphere, then
project that projection onto the 2-D plane using stereographic
projection
-To visualize a complex 4-D object in 3-D, first project the
planes and vertices of the object onto a 4-D hypersphere, then
project that projection onto the 3-D hyperplane (i.e. 3-D
space) using stereographic projection
Stereographic projection of 4-D
hypersphere onto 3-D space
-Hypersphere appears as a torus which turns inside out
(signified by the change in direction of the bands around the
torus)
3-D stereographic projection of
4-D hypercube
3-D stereographic projection of 4-D
hyperdodecahedron (120-cell)
- This hecatonicosachoron has 120 dodecahedra; or 720
pentagons, 600 vertices, and 1200 edges
Another stereographic projection view
of the 120-cell
Regular polytopes in 3-D space
Tetrahedron
cube
Faces
4
triangles
6
squares
8
triangles
12
pentagons
20
triangles
Edges
6
12
12
30
30
Vertices 4
8
6
20
12
octahedron
dodecahedron icosahedron
Regular polytopes in 4-D space
5-cell
(hypertetrahedron)
8-cell
(hypercube)
16-cell
(hyperoctahedron)
24-cell
120-cell
(hyperdodecahedron)
600-cell
(hypericosahedron)
Facets
5
tetrahedrons
8
cubes
16
tetrahedrons
24
octahedrons
120
dodecahedrons
600
tetrahedrons
Edges
10
32
24
96
1200
720
5
16
8
24
600
120
Vertices
3-D stereographic projection of 4-D 5-cell
(hyper-tetrahedron)
3-D perspective projection of 4-D 5-cell
(hyper-tetrahedron)
3-D stereographic projection of 4-D 8-cell
(hyper-cube)
3-D perspective projection of 4-D 8-cell
(hyper-cube)
3-D stereographic projection of 4-D 16-cell
(hyper-octahedron)
3-D perspective projection of 4-D 16-cell
(hyper-octahedron)
3-D stereographic projection of 4-D 24-cell
3-D perspective projection of 4-D 24-cell
3-D stereographic projection of 4-D 120-cell
(hyper-dodecahedron)
3-D perspective projection of 4-D 120-cell
(hyper-dodecahedron)
3-D stereographic projection of 4-D 600-cell
(hyper-icosahedron)
3-D perspective projection of 4-D 600-cell
(hyper-icosahedron)
Sequence of 3-D cross-sections of
4-D 600-cell (hyper-icosahedron)
- Each vertex is shared by 20 tetrahedra
clustered around it; thus, when the shape is cut
near a vertex, the 20 tetrahedra are cut through
equally, producing an icosahedron
- As we get closer to the center the icosahedrons
get bigger
- Here the cross-section begins to become a
stellated dodecahedron (12-sided figure)
Sequence of 3-D cross-sections of
4-D 600-cell (hyper-icosahedron)
- On each of the 12 pentagonal sides of the
dodecahedron in this cross-section there is a set
of 5 triangles which bulge out slightly
- The colors indicate where the cross-sectional
cut is in relation to our position in 4-D space; red
is farthest away and blue is closest
- In these sections we can see that at each (1-D)
edge of the 600-cell there are 5 tetrahedra
Sequence of 3-D cross-sections of
4-D 600-cell (hyper-icosahedron)
- In this section ten narrow triangles meet at each
vertex of an icosahedron
- This section represents the middle of the 600-cell,
so it is the largest cross-section, having 12
pentagonal clusters of faces centered at the
vertices of an icosahedron
Sequence of 3-D cross-sections of
4-D 600-cell (hyper-icosahedron)
- In these sections the entire sequence reverses,
because we are moving out of the 600-cell
toward the “front”
Sequence of 3-D cross-sections of
4-D 600-cell (hyper-icosahedron)
Tessellation of 3-D space by cubic cells
Tesseractic tessellation of 4-D space
(3-D perspective projection)
- The unit cell is the hypercube (4-D 8-cell)
Demitesseractic tessellation of 4-D space
(3-D perspective projection)
- The unit cell is the 16-cell, which is the 4-D analogue of
the 3-D octahedron
Icositetrachoronic tessellation of 4-D space
(3-D stereographic projection)
- The unit cell is the 24-cell, which has no analogue among
the 3-D regular polytopes
Space inside a sphere of various
dimensions
- The space “inside” a 1-D “sphere” (line) of radius r is the
length 2r
- The space inside a 2-D “sphere” (circle) of radius r is the
area pr2
- The space inside a 3-D sphere of radius r is the volume 4/3pr3
- The space inside a 4-D hypersphere of radius r is the
hypervolume 1/2p2r4