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Visualizing 3D Projections of Higher Dimensional Polytopes:
An Approach Linking Art and Computers
Yaxal Arenas1, Ricardo Pérez-Aguila2
Universidad de las Américas, Puebla (UDLAP)
2
Departamento de Computación, Electrónica, Física e Innovación; Departamento de Actuaría y Matemáticas
Ex-Hacienda Santa Catarina Mártir, Puebla, 72820, México
[email protected], [email protected]
1
Abstract – Visualizing higher dimensional polytopes
through computer graphics offers a way to
understand and to analyze these interesting objects.
Such advantages can be improved by considering
polytopes’ three-dimensional projections which are
embedded in our space. In this work we will describe
basic methodologies for polytopes’ visualization and
the way such procedures assist the creation of
sculptures that represent 3D projections of polytopes.
Such sculptures produce a distinct view of what is
seen in the computer screen.
Keywords – Euclidean Higher Dimensional Spaces,
Polytopes Visualization, Computer Graphics & Art.
I. INTRODUCTION
Coxeter [4] defines polytope as the general term of
the sequence "point, line segment, polygon, polyhedron, ...," or more specifically, as a finite region of
n-dimensional space enclosed by a finite number of
hyperplanes. Is it possible to visualize polytopes
embedded in spaces with more than three
dimensions? The task of visualizing polytopes in the
fourth and higher dimensional spaces has been
widely boarded from the perspective of the computer
graphics field. However, we could find restrictive the
visualization of polytopes on a 2D computer screen
in such a manner that having an extra dimension in
the visualization could help us to analyze and
understand their relations and properties in a more
profound and didactical way. In this sense we can
appeal to incorporate some techniques in order to
have a more realistic 3D projection.
At this point it is interesting to consider the relation
between computers and art with the objective of
visualizing “3D shadows” of polytopes. If these
“shadows” are embedded in our 3D world, then we
can navigate around them (or inside them, why not?)
with the advantage that we can appreciate properties
and phenomena associated to these projected
higher dimensional polytopes. In this work we will
describe some aspects related to some four and five
dimensional polytopes and we will exemplify the
mappings from the computer screen to sculptures
which represent 3D projections of these interesting
objects. The artist, in this case the first author, modeled her sculptures starting from data and computer visualization generated by the second author.
This work is organized as follows: Section 2
describes the Bragdon sequence as a way for gene-
rating the nD hypercube. The section 3 describes
the geometrical transformations that provide the way
the artist and the computer can project higher
dimensional polytopes onto three and two
dimensional spaces respectively. Finally, the section
4 briefly describes the visualization of polytopes
through their unravelings. Along sections 3 and 4 we
will show some resulting sculptures which provide a
distinct view of what is seen in the computer screen.
II. THE HYPERCUBE IN THE nD SPACE
In [11] is presented the Claude Bragdon's method to
define a series of figures which are called the
parallelotopes [4]. Now, we proceed to describe it.
O
X
O
Figure 1. Generation and final 1D unit segment.
First a 0D point is taken and moved one unit to the
right. The path between the first and the second new
point produces a 1D segment. The first dimension,
represented by the X-axis, has appeared (Figure 1).
Y
X
O
X
O
Figure 2. Generation and final 2D unit square.
The new segment is then moved one unit upward.
The path between the first and the second new
segment produces a 2D square (a parallelogram).
The second dimension, represented by the Y-axis,
has appeared (Figure 2).
Y
Y
Z
O
X
O
X
Figure 3. Generation and final 3D unit cube.
The new square is then moved one unit forward out
this paper. The path between the first and the
second new square produces a 3D cube (a parallelepiped). The third dimension, represented by the
Z-axis, has appeared (Figure 3). Because we are
working over a 2D surface (this paper or the computer’s screen), a diagonal between X and Y-axis represents the Z-axis, however it should be interpreted
as a line perpendicular to this 2D surface.
Y
Y
Z
Figure 6. Projecting a cube onto a plane.
Z
W
O
X
O
X
Figure 4. Generation and final 4D unit hypercube.
We know that the fourth dimension has a direction
perpendicular to the other three dimensions; in this
case the W-axis is presented as a perpendicular line
to the Z-axis. Then the cube is moved one unit in
direction of the W-axis. The path (six cubes
perpendicular to the first one) between the first and
the second new cube produces the 3D boundary of
a 4D hypercube (a 4D parallelotope). The fourth
dimension has appeared (Figure 4).
Hilbert [6] determined that a hypercube is composed
of sixteen vertices, twenty-four faces and eight bounding cubes (also called cells or volumes). Similarly,
and as shown in Figure 5, all these volumes can be
grouped into four pairs of parallel cubes, moreover,
their supporting hyperplanes define two 3D spaces
parallel to each other [4]. Moreover, Coxeter [4]
points that each face is shared by two cubes not in
the same 3D space, because they form a right angle
through a rotation around the shared face's
supporting plane. These properties are visible
through Bragdon's projection (Figure 4).
When the center of projection is at the infinite then
the projection rays are parallel between them. This
projection is defined as 3D-2D parallel projection,
which informally is just to remove the Z coordinate
from the object's points if the projection plane is
z = 0 (which is the most popularly used in the
Computer Graphics field):
( x, y , z )
( x, y )
When the center of projection is on Z axis at a
distance pz from the origin, and the projection plane
is z = 0, then we have a 3D-2D perspective
projection defined as:
( x, y , z )
 x ⋅ pz y ⋅ pz 
,


 pz − z pz − z 
Banks [3] establishes that the same techniques used
to project 3D objects onto 2D planes can be applied
to project 4D polytopes onto 3D hyperplanes (our 3D
space for example). Then we have that a 4D-3D
parallel projection, which informally is the X, Y, Z
or W coordinate’s removal from the polytope's
points. It has the following definition (for the typically
removed W coordinate) [10]:
( x, y, z , w)
( x, y , z )
And a 4D-3D perspective projection is defined
when the center of projection is on W axis at a
distance pw from the origin. If the projection
hyperplane is w = 0 then we have [10]
 x ⋅ pw y ⋅ pw z ⋅ pw 
( x, y, z , w) 
,
,

 pw − w pw − w pw − w 
Figure 5. Viewing the hypercube’s boundary volumes.
III. POLYTOPES’ PROJECTION
We can define a 3D-2D projection as the
transformation of 3D scenes onto 2D viewing planes
(a computer screen for example). A projection
imitates the process by which the eye maps world
scenes into images onto the retina. In general terms,
a projection transforms points in a nD space to
points onto a lower dimensional space [12].
The projection of 3D objects is defined by projection
straight rays, which emanate from a center of
projection to pass by each point of the object and to
finally, intersect a plane and create the projection
[12] (Figure 6).
Because a 4D-3D projection will produce a volume
as the "shadow" of a 4D polytope, Hollasch [7]
considers valid to process this volume with some of
the 3D-2D projections (parallel or perspective) to be
projected finally onto a computer screen. Then we
have four possible 4D-3D-2D projections:
•
•
•
•
4D-3D Perspective Projection/3D-2D Perspective Projection
4D-3D Perspective Projection/3D-2D Parallel Projection
4D-3D Parallel Projection/3D-2D Perspective Projection
4D-3D Parallel Projection/3D-2D Parallel Projection
In the literature it is common to introduce the 4D
hypercube’s projection as a cube inside another
cube, or in other words, its central projection (Figure
7). This visualization is commonly the result of
applying the combination of 4D-3D perspective and
3D-2D perspective projections.
Figure 7. Hypercube's central projection onto 3D space.
The projection’s procedures used in 3D and 4D
spaces can be generalized for any number of dimensions such that a nD polytope is projected onto a
(n-1)D hyperplane, therefore, we have a nD–(n-1)D
projection. For visualizing a nD polytope on a computer screen, for example, the projections must be
repetitively applied, in other words, to consider projections (n-1)D–(n-2)D, (n-2)D–(n-3)D,…, 3D-2D. Finally, a 2D object will be obtained, which represents
the successive projections of the nD polytope [9].
The Parallel Projection of a nD polytope onto a
(n-1)D hyperplane, or in other words, the
nD – (n-1)D Parallel Projection consists on just
removing the n-th coordinate, whose corresponding
axis is Xn, from the nD polytope’s points [10]:
( x1 , x2 , x3 ,..., xn −1 , xn )
( x1 , x2 , x3 ,..., xn −1 )
Finally, the Perspective Projection nD – (n-1)D is
defined when the projection’s center is on the
Xn-axis (which corresponds to the n-th coordinate) to
a distance pn from the origin. If the projection’s
(n-1)D hyperplane is Xn = 0, then we will have the
definition [10]:
( x1 , x2 , x3 ,..., xn −1 , xn )
 x1 ⋅ pn x2 ⋅ pn x3 ⋅ pn
x ⋅ pn 
,
,
,..., n −1


pn − xn 
 pn − xn pn − xn pn − xn
Through the generalization of the parallel and perspective projections it is possible to obtain in a simple
way, the required transformations for visualizing
polytopes beyond the 4D space. For example, in
Figure 8 is presented a 5D Hypercube’s projection,
which has its center at the origin. Also, all the
projections, applied to it, (5D-4D, 4D-3D and 3D-2D)
were perspective projections. As can be seen, that
projection results to be the 5D hypercube’s central
projection, which can be considered as a 4D hypercube inside another 4D hypercube (the interior 4D
hypercube is remarked to facilitate the visualization).
Figure 8. The 5D Hypercube’s central projection.
The Table 1 shows some snapshots of a sculpture
that corresponds to the central projection, onto our
3D space, of a 5D hypercube. Snapshots a), b), and
c) correspond to aerial visualizations. In d) we have
a view of the interior 4D hypercube. Snapshot e)
shows how a set of edges are collapsed in a point
(the center of the picture). And finally, in f), we have
that a vertex in the 5D hypercube has five
perpendicular incident edges; however, due to an
effect of the 5D-4D-3D perspective projection, each
vertex in the sculpture has edges with only four
distinct directions.
n
The nD space has 2 hyper-octants: 4 quadrants in
2D space, 8 octants in 3D space, and 16
hyper-octants in 4D space. We can define an array
of 4D hypercubes by positioning each one in each
hyper-octant. Under such positioning we have that
these 16 hypercubes have one of their vertices at
the origin of 4D space. In any of these hypercubes
Table 1. Three-Dimensional Projection of a 5D Hypercube: Sculpture in steel by Yaxal Arenas (52 cm x 52 cm x 52 cm)
a)
b)
c)
d)
e)
f)
we have that the four edges incident to the origin are
coincident with the four coordinate axes (see Figure
9). Moreover, in the array we can find that in some
pairs of hypercubes, a kD boundary element,
1 ≤ k < 4, can be shared. That is, two hypercubes
can share an edge (1D), a face (2D), or one of its
boundary volumes (3D). Figure 9 shows all the
adjacencies between the hypercubes in our array.
The Table 2 shows some snapshots of a sculpture
that corresponds to a projection, onto our 3D space,
of the array of sixteen hypercubes previously
described. In snapshots a), b), c) and d), the aerial
views show how the selected projection makes
clearly visible three sets of eight cubes (painted in
blue, yellow and red). The eight cubes in each set
share a common vertex. There are another nine sets
of eight cubes which are deformed by the selected
projection. Snapshots e) and f) are closer views
which show how each vertex in the sculpture has
incident edges in four distinct directions.
IV. UNRAVELING POLYTOPES
A cube can be unraveled as a 2D cross. The six
faces on the cube's boundary will compose the 2D
cross (Figure 10.a). The set of unraveled faces is
called the unravelings of the cube.
In analogous way, a hypercube also can be
unraveled as a 3D cross. The 3D cross is composed
by the eight cubes that form the hypercube's
boundary [8]. This 3D cross was named tesseract by
C. H. Hinton (Figure 10.b).
a)
a)
b)
Figure 10. a) Unraveling the cube.
b) The unraveled hypercube (the tesseract).
b)
Figure 9. 2D and 3D projections (a and b respectively) of an
array of sixteen 4D hypercubes. Sculpture by Yaxal Arenas
(40 cm x 40 cm x 40 cm).
A flatlander will visualize the 2D cross, but he will not
be able to assembly it back as a cube (even if the
specific instructions are provided). This fact is true
because of the needed face-rotations in the third
dimension around an axis which are physically
impossible in the 2D space. However, it is possible
for the flatlander to visualize the raveling process through the projection of the faces and their
Table 2. Three-Dimensional Projection of a configuration of sixteen 4D hypercubes incident to a common vertex:
Sculpture in steel by Yaxal Arenas (50 cm x 40 cm x 38 cm).
a)
b)
c)
d)
e)
f)
movements onto the 2D space where he lives [1].
Analogously, we can visualize the tesseract but we
will not be able to assembly it back as a hypercube.
We know this because of the needed volumerotations in the fourth dimension around a plane
which are physically impossible in our 3D space [1].
However, it is possible for us to visualize the raveling
process through the projection of the volumes and
their movements onto our 3D universe.
The Table 3 presents some snapshots from the
cube's unraveling sequence. The blue face is
embedded in the target 2D space; hence, it does not
require to be manipulated in the unraveling process.
In snapshots 1 and 2, the applied rotations are 0°
and ±30° (the rotation’s sign depends of the
analyzed face). In snapshot 3, the applied rotation is
±53°. In snapshot 4 the applied rotation is ±90°; the
faces, adjacent to the immobile face, have finished
their movements. In snapshots 5 to 6, the red face
moves independently and the applied rotations are
+60° and +90° respectively.
Observing the unravelings for a square (a 2D cube),
a cube and the 4D hypercube and the fact a nD
parallelotopes-family share analogous properties [4]
we can generalize the nD hyper-tesseract (n≥1) as
the result of the (n+1)-D parallelotope’s unraveling
with the following properties [1]:
• The (n+1)-D hypercube will have 2(n+1) nD cells
on its boundary [2].
• A central cell is static in the unraveling process.
• 2(n+1)-2 cells are adjacent to central cell. All of
them will share a (n-1)-D cell with central cell.
• A satellite cell will not be adjacent to central cell
because their supporting hyperplanes are parallel.
It will share a (n-1)-D cell with the selected
adjacent cell.
• All the adjacent cells and satellite cell during the
unraveling process will rotate ±90° around the
supporting hyperplane of the (n-1)-D shared cells.
Table 4. Unraveling the hypercube (satellite volume is
shown in blue and immobile volume in red).
Table 3. Unraveling the cube.
1
4
2
5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
3
6
The Aguilera & Pérez method [1] provides a
methodology for unraveling the hypercube and
getting the 3D-cross (tesseract) that corresponds to
the hyper-flattening of their boundary. The
transformations to apply include rotations around a
plane (See [5] for details about the topic of rotations
in nD space). In [1], specific details about the
procedure can be found. In fact, the Aguilera &
Pérez method can be used, by taking in account the
specific aspects, for unraveling nD hypercubes.
The Table 4 presents some snapshots from the 4D
hypercube's unraveling sequence which was
obtained by means of the Aguilera & Pérez method.
The red volume is embedded in the target 3D space;
hence, it does not require to be manipulated in the
unraveling process. In snapshots 1 to 6, the applied
rotations are ±0°, ±15°, ±30°, ±45°, ±60° and ±75°
(the rotation’s sign depends on the analyzed
volume). In snapshot 7, the applied rotation is ±82°;
the blue volume looks like a plane --an effect due to
the selected 4D-3D projection [1]. In snapshot 8, the
applied rotation is ±90°; the volumes, adjacent to the
immobile volume, finish their movements. In
snapshots 9 to 14 the blue volume, called in [1]
satellite volume, moves independently and applied
rotations are +15°, +30°, +45°, +60°, +75° and +90°.
For example, the 4D hyper-tesseract is the result of
the 5D hypercube’s unraveling. See Figure 11.a. The
4D hyper-tesseract will be composed by 10 4D
hypercubes, where one of them will be embedded in
the target 4D space, eight of them are adjacent to
red hypercube (they share a volume) and the last
one will be a blue hypercube which shares a volume
with any hypercube except that embedded in the
target 4D space. Mobile 4D hypercubes will rotate
around a 3D hyperplane during the unraveling
process. Figure 11.b shows a 4D-3D-2D perspective
projection of the assembled 4D hyper-tesseract.
-Z
proposed to directives of the UDLAP’s Actuarial
Sciences and Mathematics Department. The objective behind that proposed course is to introduce to
students, in the areas of Mathematics, Physics and
Computer Science, to the visualization of hyperdimensional objects by means of the computer. In
order to express the properties associated to these
polytopes we expect to use as didactical utility the
3D sculptures we have presented in this work. In this
point it is important to mention the support and comments provided by Guillermo Romero-Meléndez,
PhD, and Antonio Aguilera, PhD. Finally we mention
that in the last National Congress of the Mexican
Mathematical Society (October 2006), we presented
a talk about the relationship between computers and
art under the visualization of nD polytopes. Such
presentation generated a good interest in the topic.
X
Y
W
-Y
-X
Z
a)
b)
Figure 11. a) The possible adjacency relations between
the 4D hypercubes that compose the 4D hyper-tesseract.
b) A 4D-3D-2D perspective projection of the
4D hyper-tesseract: W-axis collapses in a point due to an
effect of the projection.
REFERENCES
The Table 5 shows a sculpture that corresponds to a
3D projection of the 4D hyper-tesseract. Snapshot a)
shows a perspective of the sculpture which resembles that the composing 4D hypercubes have the
Bragdon’s projection. In snapshot b) we have another view of the sculpture where some hypercubes
can be appreciated in central projection. In snapshot
c) is shown an aerial view which resembles the projection from Figure 11.b. Finally, in snapshot d), a
close view reveals how each vertex in the projection
has incident edges in four distinct directions.
[1]
Aguilera Ramírez, A. & Pérez Aguila, R. A Method For
Obtaining The Tesseract By Unraveling The 4D Hypercube.
Journal of WSCG 2002. Vol. 10, Number. 1, pp. 1-8.
February 4-8 2002. Plzen, Czech Republic.
[2] Banchoff, T.F. Beyond the Third Dimension. Scientific
American Library, 1996.
[3] Banks, David. Interactive Manipulation and Display of TwoDimensional Surfaces in Four-Dimensional Space. Proceedings of the 21st annual conference on Computer graphics,
July 24 - 29, 1994, Orlando, FL USA, pp. 327 - 334.
[4] Coxeter, H.S.M. Regular Polytopes. Dover Publications,
Inc., New York, 1963.
[5] Duffin, Kirk & Barnett, William. Spiders: A new user
interface for rotation and visualization of n-dimensional
points sets. Proceedings of the 1994 IEEE Conference on
Scientific Visualization.
[6] Hilbert, D. & Cohn-Vossen, S. Geometry and the
Imagination. Chelsea Publishing Company, 1952.
[7] Hollasch, S.R. Four-Space Visualization of 4D Objects. MSc
Thesis. Arizona State University, 1991.
[8] Kaku, M. Hyperspace: A Scientific Odyssey Through
Parallel Universes, Time Warps, and the Tenth Dimension.
Oxford University Press, 1994.
[9] Noll, A. Michael. A Computer Technique for Displaying
n-Dimensional Hyperobjects. Communications of the ACM,
Volume 10, Number 8, pp. 469-473.
[10] Pérez Aguila, R. The Extreme Vertices Model in the 4D
space and its Applications in the Visualization and Analysis
of Multidimensional Data Under the Context of a Geographical Information System. MSc Thesis. Universidad de las
Américas, Puebla. Puebla, México, May 2003.
[11] Rucker, R.V.B. Geometry, Relativity and the Fourth
Dimension. Dover Publications, Inc., New York, 1977.
[12] Foley, Van Dam, Feiner, Hughes & Phillips. Introducción a
la graficación por computador. Addison-Wesley, 1996.
V. CONCLUSIONS AND FUTURE WORK
Computer generated visualization of polytopes is
very fruitful in generating the curiosity in students for
examining these interesting objects and such
curiosity can be positively increased by showing
them 3D tangible projections.
Currently, at the UDLAP, we have used successfully
the focusing we have described in this work in
several activities. In April 2006, in a set of
presentations entitled “Day of Art and Science”,
which was organized by the Student Council of the
UDLA (CEUDLA), we presented the computed
generated projections of 4D and 5D polytopes and
some of their associated 3D projections.
At the moment, an undergraduate course entitled
“Visualizing the n-Dimensional Space” has been
Table 5. Three-Dimensional Projection of the 4D hyper-tesseract: Sculpture in steel by Yaxal Arenas (65 cm x 65 cm x 80 cm).
a)
b)
c)
d)