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]

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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

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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)