A SYMMETRIC THREE-DIMENSIONAL MODEL OF THE HYPERCUBE

Transcription

1 Symmetry: Culture and Science Vol. 16, No.1, page_first-page_last, 2005 A SYMMETRIC THREE-DIMENSIONAL MODEL OF THE HYPERCUBE László Vörös Architect, b. Mohács, Hungary, Address: Institute of Architecture, M. Pollack Technical Faculty, University of Pécs, Boszorkány utca 2., 7624 Pécs, Hungary vorosl@witch.pmmf.hu Fields of interest: Descriptive geometry, CAD, architecture Publications: Vörös, L., (2005) Reguläre Körper und mehrdimensionale Würfel, KoG Scientific-Professional Journal of Croatian Society for Constructive Geometry and Computer Graphics, no.9, Abstract: Lift the outer endpoints of each radius of a regular k-sided polygon to an angle identical to the plane of the polygon. These sections are the edges conjoining in the starting vertex of the 3-dimensional model of the k-cube. The model s construction with the well known sliding-method creates a lattice structure, whose outer bars equal a zonotope s edges. This structure keeps the 3-cube s central symmetry and rotational symmetry related to the main diagonal referred to the groups of any j<k dimensioned element. It is also possible to get to the endpoint of the main diagonal from the starting point along easily recognizable bar-chains, whose binding points join on one helix each. The common lead of these helixes is the main diagonal. Increasing the number of the sections in the bar-chain infinitely, a continuous helix is created, whose polar distribution means the rotation around the lead. Therefore the shell of the n-cube s 3-model is generated as the surface of a solid of rotation in which any k- dimensional cube s 3-model can be constructed. According to these our lattice 3- model of any k-cube can also be generated either as ray-groups of the edges or as sequences of bar-chains originated from a separate helix. Keywords: hypercube, 3-dimensional construct, symmetry. Remark: The topic requires transparent and coloured figures. They are in the text denoted but can be found with a more detailed text on:

2 2 LÁSZLÓ VÖRÖS Lift the outer endpoints of each radius of a regular k-sided polygon to an angle identical to the plane of the polygon. These sections are the edges conjoining in the starting vertex of the 3-dimensional model of the k-cube (further: 3-model) [2],[4]. The model s construction with the well known sliding-method creates a lattice structure, whose outer bars equal a zonotope s [3] edges. This structure keeps the 3-cube s central symmetry and rotational symmetry related to the diagonal joining the starting vertex referred to the groups of any j<k dimensioned element. This diagonal is further on related to as main diagonal (fig. 1). The number of the vertices ( j = 0) of the k-cube is 2 k. In case of further values of j the number of the elements is 2 k-j C k ; j. (C: combination without repetition.) It is also possible to get to the endpoint of the main diagonal from the starting point along easily recognizable bar-chains, whose binding points (the outer vertices of the model) join on one helix each. The common lead of these helixes is the main diagonal. According to these attributes, such a bar-chain can, distributed around the main diagonal in the number equal to the number of his elements and the given distribution mirrored to the centre point of the common lead, generate the outer edges of the k-cube s 3-model (fig. 2). Increasing the number of the sections in the bar-chain infinitely a continuous helix is created, whose polar distribution means, according to the procedure so far, the rotation around the lead, therefore the shell of the n-cube s 3-model is generated as the surface of a solid of rotation in which any k-dimensional cube s 3-model can be constructed. The surface of revolution is generated as an infinite amount of central or plansymmetrical double helixes, similar to the straights appearing on the surface of a hyperbole rotated around the imaginary axis. According to the construction rules so far the full 3-model of the n-cube generates a solid of rotation, which is optionally expandable in direction of the rotation axis (fig. 3). Thus the height of the solid is also determinable, as it holds the 3-cubes 3-model with the proportions of the normal cube (fig. 4). Interpreting the construction of the k-cubes 3-model as a sequence of dispositions, the increasing dimensioned cubes 3-models can be easily separated, for example starting from the upper end of the main diagonal. In this model of the k-cube, the edges of the 0,1, k cube elements model sequence are parallel to the k elements of the bar-chain approaching the starting helix, and the 1,2, k dispositions vectors are following one another along this bar-chain (fig. 5). Therefore the model s 0,1, k-1 segments can also be interpreted as intersections of two full models each (fig. 6), and the equal

3 A SYMMETRIC THREE-DIMENSIONAL MODEL OF THE HYPERCUBE 3 dimensioned parts are positioned around the main diagonal of the model, symmetrical to its centre point. According to these our model can also be generated as described in [4]. In this procedure the vertices of the bar-chain replacing the helix will be referred to as C 0 -C k, its segments as a 1 -a k, their disposed copies as b 1 -b k and the k-fold distributed C vertices as D. This algorithm is difficult to follow, even by using a CAD-program, not only in case of higher dimensions, but also for the complexity of handling the coinciding and masking elements. A sequence of the construction is represented by figures 7.a-c. The polygons were constructed with k vertices defined by the b-chains upper endpoints on the D points level-planes, to be able to indicate the models inner vertices by the polygons vertices. This is also a complex procedure, but it gives a descriptive picture of the polygons proportions to the starting helix and one another. These polygons are defined by the outer vertices (fig. 8). Finding the inner vertices is made easier, if we know how many vertices there are in the k-folded polar distribution on each level to be looked for (fig. 9). Because the vertices are following a k-folded polar distribution on level k+1, and there are always k edges conjoining in one vertex, it can be pointed out, that there are k edges starting from the vertex on the main diagonal s endpoint, and there are also k edges starting from their endpoints, still, only k-1 of these are reaching the next level. Two edges are ending in each vertex of the level, and k-2 are leading from it, etc. This tendency gets opposed in middle-height by the model s symmetry. On behalf of the k-folded polarity, the points equal to the remainder of the division of the number of vertices on a given level by k, coincide in the centre point of the level. The multiple coinciding of the inner vertices polar groups is also possible (fig. 9). Following the arrangement of edges and vertices as described above, a new algorithm is possible to determine to create the model. No edge may start as continuing to the next level from the edges arriving in a given vertex of a level. It is enough to define these illegal edges in one vertex of each polar group, because they are also arranged circularly symmetrical. The usable edges continuing from a given vertex arrive in one vertex of the next levels polar groups each (fig. 9). Summarizing all our experience so far, a symmetric 3-model of the k-cube can be described originated from a separate helix as it follows (fig. 10):

4 4 LÁSZLÓ VÖRÖS - replacing the helix with a bar-chain of k sections of identical length, whose vertices are joining the helix - the sections of the bar-chain replacing the helix are designated as a 1 -a k, their disposed copies as b 1 -b 2 - generating the bar-chains A 3 (b 3,b 4,... b k ), A 4 (b 4,b 5,... b k ),... A k (b k ) - generating the groups of bar-chains B 1 (A 3,A 4, A k ), B 2 (A 4,A 5, A k ), B k-3 (A k-1,a k ) by moving the bar-chains into the suitable common starting point, whose disposed copies will later be called C 1 -C k-3. - creating C 1 in the upper endpoint of a 1 - creating C 2 in the upper endpoint of a 2 - creating C 3 in the upper endpoints of a 3 and all b 3 segment of all so far created C creating C k-3 in the upper endpoints of a k-3 and all b k-3 segment of all so far created C - distributing all a and C k-folded around the starting helix s lead - mirroring the copies of all created elements to the plane halving the lead and being perpendicular to it - (rotating the mirrored elements 180º around the lead) - eliminating the coinciding elements, and give the edges the colour of the starting bar-chain s elements parallel to them.

5 A SYMMETRIC THREE-DIMENSIONAL MODEL OF THE HYPERCUBE 5 The creation of the constructions and figures required for the paper was aided by the AutoCAD program and the self developed Autolisp routines. REFERENCES [1] Miyazaki, K., Adventure in Multidimensional Space: The Art and Geometry of Polygons, Polyhedra and Polytopes, New York: Wiley, 1986 [2] Vörös, L. (2005) Reguläre Körper und mehrdimensionale Würfel, KoG Scientific-Professional Journal of Croatian Society for Constructive Geometry and Computer Graphics, no.9, [3] [4]

6 László Vörös University of Pécs, Pollack M. Faculty of Engineering, Institute of Architecture A Symmetric 3-Dimensional Model of the Hypercube Lift the outer endpoints of each radius of a regular k-sided polygon to an angle identical to the plane of the polygon. These sections are the edges conjoining in the starting vertex of the 3-dimensional model of the k-cube (further: 3-model) [2],[4]. The model s construction with the well known sliding-method creates a lattice structure, whose outer bars equal a zonotope s [3] edges. This structure keeps the 3-cube s central symmetry and rotational symmetry related to the diagonal joining the starting vertex referred to the groups of any j<k dimensioned element. This diagonal is further on related to as main diagonal (fig. 1). The number of the vertices ( j = 0) of the k-cube is 2 k. In case of further values of j the number of the elements is 2 k-j C k ; j. (C: combination without repetition) Figure 1 It is also possible to get to the endpoint of the main diagonal from the starting point along easily recognizable barchains, whose binding points (the outer vertices of the model) join on one helix each. The common lead of these helixes is the main diagonal. According to these attributes, such a bar-chain can, distributed around the main diagonal in the number equal to the number of his elements and the given distribution mirrored to the centre point of the common lead (otherwise mirrored to the plane halving the main diagonal and being perpendicular to it, and rotating the mirrored parts 180º around the main diagonal), generate the outer edges of the k-cube s 3-model. Obviously the order of the mirroring and the circular distribution can be changed (fig. 2). Another way is to mirror the first barchain to the plane of its beginning part and lead, then make a polar distribution of the pair of bar-chains sym-metrical to itself. 1

7 Figure 2 Figure 3 2

8 Increasing the number of the sections in the bar-chains infinitely a continuous helix is created, whose polar distribution means, according to the procedure so far, the rotation around the lead, therefore the shell of the n-cube s 3-model is generated as the surface of a solid of rotation in which any k-dimensional cube s 3-model can be constructed. The surface of revolution is generated as an infinite amount of central or plan-symmetrical double helixes, similar to the straights appearing on the surface of a hyperbole rotated around the imaginary axis. According to the construction rules so far the full 3-model of the n-cube generates a solid of rotation, which is optionally expandable in direction of the rotation axis (fig. 3). Thus the height of the solid is also determinable, as it holds the 3-cubes 3-model with the proportions of the normal cube. Move the 3-cubes diagonal perpendicular to the base plane, and project it orthogonally onto the plane! The projected regular hexagon consists of six regular triangles. The circles constructed around these triangles, are the base circles of six cylinders, and the common generant of these cylinders is the diagonal of the cube. Therefore common leaded leftand right threaded helixes can be constructed onto the cylinders, on which the cube s three vertices are joining each, i.e. 3 edges of the cube are approaching the helixes (fig. 4). In case of an optional k-cube, the diameter of the cylinder of the helix and its lead remains unchanged, and the helixes are replaced with bar-chains of k elements. Utilizing k-fold polar distribution around the lead, the k-cube s afore mentioned 3-model s outer bars can be generated (fig. 2). Figure 4 3

9 Figure 5 Figure 6 4

10 Interpreting the construction of the k-cubes 3-model as a sequence of dispositions, the increasing dimensioned cubes 3-models can be easily separated, for example starting from the upper end of the main diagonal. In this model of the k-cube, the edges of the 0,1, k cube elements model sequence are parallel to the k elements of the bar-chain approaching the starting helix, and the 1,2, k dispositions vectors are following one another along this bar-chain (fig. 5). Therefore the model s 0,1, k-1 segments can also be interpreted as intersections of two full models each (fig. 6), and the equal dimensioned parts are positioned around the main diagonal of the model, symmetrical to its centre point. According to these our model can also be generated as described here: - the vertices of the bar-chain replacing the helix will be referred to as C 0 -C k, its segments as a 1 -a k, and their disposed copies as b 1 -b k. - distributing the vertices of the bar-chain k-folded. These will be uniformly referred to as D - (moving the copy of a k to C 1 along the bar-chain, starting from C k, thus generating b k ) o (distributing b k around C 1 k-folded) o (erasing those distributed elements whose endpoints do not join D points both) - generating chain (b k, b k-1 ) in C 2 o (k-fold distributing the bar-chains created above C 1 so far and leading upwards from it around C 1 ) o k-fold distributing the bar-chains created above C 2 so far and leading upwards from it around C 2 o erasing those distributed bar-chains whose endpoints do not join D points both - generating chain (b k,, b k-j+1 ) in C 3 o (k-fold distributing the bar-chains created above C 1 so far and leading upwards from it around C 1 ) o o k-fold distributing the bar-chains created above C j so far and leading upwards from it around C j o erasing those distributed bar-chains whose endpoints do not join D points both - - generating chain (b k,, b 2 ) in C k-1 o (k-fold distributing the bar-chains created above C 1 so far and leading upwards from it around C 1 ) o o k-fold distributing the bar-chains created above C k-1 so far and leading upwards from it around C k-1 o erasing those distributed bar-chains whose endpoints do not join D points both - mirroring the copies of the created elements to the plane perpendicular to the lead and halving it - rotating the mirrored elements 180º around the lead Figure 7.a 5

11 Figure 7.b Figure 7.c This algorithm is difficult to follow, even by using a CAD-program, not only in case of higher dimensions, but also for the complexity of handling the coinciding and masking elements. A sequence of the construction is represented by figures 7.a-c. (It is more effective to discern the elements with colours instead with the indexes above, marking the element sequences with colours in spectral order. Many coinciding elements and elements to be erased afterwards can be avoided by abandoning the bars falling on the highest level.) So I have constructed the polygons with k vertices defined by the b-chains upper endpoints on the D points level planes, to be able to indicate the models inner vertices with the polygons vertices. This is also a complex procedure, but it gives a descriptive picture of the polygons proportions to the starting helix and one another. These polygons are defined by the outer vertices (fig. 8). Finding the inner vertices is made easier, if we know how many vertices there are in the k-folded polar distribution on each level to be looked for (fig. 9). 6

12 Figure 8 Figure 9 7

13 Because the vertices are following a k-folded polar distribution on level k+1, and there are always k edges conjoining in one vertex, it can be pointed out, that there are k edges starting from the vertex on the main diagonal s endpoint, and there are also k edges starting from their endpoints, still, only k-1 of these are reaching the next level. Two edges are ending in each vertex of the level, and k-2 are leading from it. This tendency gets opposed in middle-height by the model s symmetry. If k is an even number then k/2 edges are reaching the medial plane, and the same number of edges are leading from it if k is an uneven number, the number of edges reaching and leading from the two medial planes changes from (k+1)/2 to (k-1)/2 symmetrically. On behalf of the k-folded polarity, the points equal to the remainder of the division of the number of vertices on a given level by k coincide in the centre point of the level. The multiple coinciding of the inner vertices polar groups is also possible (fig. 9). Figure 9.a Figure 9.b 8

14 Figure 9.c Figure 9.d 9

15 Stretching appropriately the model, it can be obtained, that most of the outer vertices join the face of a sphere and all vertices are placed in vertices of regular polygons lain on the level-planes arranged with equilised altitude (fig. 9.a-c). We can see in figure 9.b that on the 3-model of the 5-cube the outer vertices of four levels from the six one can join the face of the sphere. Because the pentagons of the inner and outer vertices origin from each other, the horisontal arrange of the vertices follows the golden section (fig. 9.d). Following the arrangement of edges and vertices as described above, a new algorithm is possible to determine to create the model. No edge may start as continuing to the next level from the edges arriving in a given vertex of a level. It is enough to define these illegal edges in one vertex of each polar group, because they are also arranged circularly symmetrical. The usable edges continuing from a given vertex arrive in one vertex of the next levels polar groups each (fig. 9). Figure 10 Summarizing all our experience so far, a symmetric 3-model of the k-cube can be described originated from a helix as it follows (fig. 10 representing in common the last two steps): - replacing the helix with a bar-chain of k sections of identical length, whose vertices are joining the helix - the sections of the bar-chain replacing the helix are designated as a 1 -a k, their disposed copies as b 1 -b 2 - generating the bar-chains A 3 (b 3,b 4,... b k ), A 4 (b 4,b 5,... b k ),... A k (b k ) - generating the groups of bar-chains B 1 (A 3,A 4, A k ), B 2 (A 4,A 5, A k ), B k-3 (A k-1,a k ) by moving the barchains into the suitable common starting point, whose disposed copies will later be called C 1 -C k-3. - creating C 1 in the upper endpoint of a 1 - creating C 2 in the upper endpoint of a 2 - creating C 3 in the upper endpoints of a 3 and all b 3 segment of all so far created C

16 - creating C k-3 in the upper endpoints of a k-3 and all b k-3 segment of all so far created C - distributing all a and C k-folded around the starting helix s lead - mirroring the copies of all created elements to the plane halving the lead and being perpendicular to it - in case of odd k, rotating the mirrored elements 180º around the lead - eliminating the coinciding elements, and give the edges the colour of the starting bar-chain s elements parallel to them. The 3-model of the k-cube can give special arrange too. We can find in these cases other ways to construct it with bar-chains joining helices. The model of the 6-cube shows an example to this in the figure 11. Figure 11 The creation of the constructions and figures required for the paper was aided by the AutoCAD program and the self developed Autolisp routines. References: [1] Miyazaki, K., Adventure in Multidimensional Space: The Art and Geometry of Polygons, Polyhedra and Polytopes, Wiley, New York, [2] Vörös, L., Reguläre Körper und mehrdimensionale Würfel, KoG N o 9, Zagreb, [3] [4] Vörös László vorosl@witch.pmmf.hu Pécsi Tudományegyetem Pollack M. Műszaki Kar H-7624 Pécs, Boszorkány u