EECS Computer Science Division University of California, Berkeley ISAMA 2004 Bridges, Pécs, 2010 My Search for Symmetrical Embeddings of Regular Maps Title Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

Math  Art This is a “math-first” talk ! “Art” comes into it in secondary ways: The way I find my solutions is more an “art” than a science or a formal math procedure; How to make the results visible is also an “art”; Some of the resulting models can be enhanced so that they become “art-objects” on their own. This is a much more mathematical talk than the one I gave last year.

Regular Maps of Genus Zero ISAMA 2004 Regular Maps of Genus Zero Hosohedra So, what are these “regular maps” that I am going to tell you about ? They are networks of edges, vertices, and faces where all these elements are indistinguishable from their siblings. On a sphere or on any other genus-0 surface (with no holes or handles) these are the only possible regular maps: The 5 Platonic solids on the left and the infinite series of hosohedra on the right – with just 2 vertices at the two poles, and a series of N regularly spaced edges between them – thus forming a set of orange slices… -- and their duals, the dihedra, with just two faces and a regular n-gon around the equator. Platonic Solids Di-hedra

The Symmetry of a Regular Map ISAMA 2004 The Symmetry of a Regular Map The key operation that proves that one of these networks is regular is: Take an arbitrary edge and move it onto any other arbitrary edge (in either of 2 orientations), then the whole network must be able to settle down, coinciding with itself, -- with every edge finding a matching edge, and every vertex or face finding a matching vertex or face. After an arbitrary edge-to-edge move, every edge can find a matching edge; the whole network coincides with itself.

On Higher-Genus Surfaces: only “Topological” Symmetries ISAMA 2004 On Higher-Genus Surfaces: only “Topological” Symmetries Edges must be able to stretch and compress On surfaces of higher genus, we can no longer expect an exact geometrical match when we move the network around. As is obvious on this torus, edges on the outside will become stretched and those on the inside will be compressed. But if we assume, the surface is covered with a thin stretchable fabric, such as nylon stockings, on which we draw our networks, then we can still use the same edge-to-edge matching criterion to find out whether we have a regular map. Infinitely many regular maps are possible on a torus: just pick a square matrix of quadrilateral tiles. But if the number of tiles around the big loop is different from the number around the ring, as shown on the right, then it is NOT a regular map, If we try to rotate a tile by 90* then the network cannot cover itself. Regular map on torus (genus = 1) NOT a regular map: different-length edge loops 90-degree rotation not possible

How Many Regular Maps on Higher-Genus Surfaces ? ISAMA 2004 How Many Regular Maps on Higher-Genus Surfaces ? Two classical examples: As we move to higher-genus surfaces, there are now more constraints. Any path around a handle or around a hole must have the same number of edges, and this is harder to achieve. Until the 1980s only a few regular maps of higher genus were known. Then Coxeter and others started to look for them, and every time they found one, it was a big event. But for quite some time, it remained unclear how many such regular maps there might exist. Here are 2 well known classical examples: The genus-2 map R2.1, having the same symmetry properties as the quaternion group; and the Klein quartic about which I talked a few years ago. R2.1_{3,8} _12 16 triangles Quaternion Group [Burnside 1911] R3.1d_{7,3} _8 24 heptagons Klein’s Quartic [Klein 1888]

zig-zag path closes after 8 moves ISAMA 2004 Nomenclature “Eight-fold Way” zig-zag path closes after 8 moves Regular map genus = 3 # in that genus-group the dual configuration heptagonal faces valence-3 vertices length of Petrie polygon:  Let’s analyze the expressions that characterize these particular regular maps. The leading “R” says this is a regular map; then we read the genus of that map; next we find a simple numerical identifier; followed by a “d” if it is the dual map. In curly parentheses we find the Schläfli symbol used to define regular polytopes and tessellations by the # of sides on each polygon and the valence of the vertices. Finally, the Petrie polygon is zig-zag-path on the edge structure. It always gets back to the starting point after a fixed number of moves, regardless where you start (if it doesn’t, it is NOT a regular map). For this particular map, it takes 8 steps, -- that is why Helaman Ferguson called his artistic sculpture celebrating this surface: the Eight-fold Way. R3.1d_{7,3}_8 Schläfli symbol

2006: Marston Conder’s List ISAMA 2004 2006: Marston Conder’s List http://www.math.auckland.ac.nz/~conder/OrientableRegularMaps101.txt Orientable regular maps of genus 2 to 101: R2.1 : Type {3,8}_12 Order 96 mV = 2 mF = 1 Defining relations for automorphism group: [ T^2, R^-3, (R * S)^2, (R * T)^2, (S * T)^2, (R * S^-3)^2 ] R2.2 : Type {4,6}_12 Order 48 mV = 3 mF = 2 Defining relations for automorphism group: [ T^2, R^4, (R * S)^2, (R * S^-1)^2, (R * T)^2, (S * T)^2, S^6 ] R2.3 : Type {4,8}_8 Order 32 mV = 8 mF = 2 Defining relations for automorphism group: [ T^2, R^4, (R * S)^2, (R * S^-1)^2, (R * T)^2, (S * T)^2, S^-2 * R^2 * S^-2 ] R2.4 : Type {5,10}_2 Order 20 mV = 10 mF = 5 Defining relations for automorphism group: [ T^2, S * R^2 * S, (R, S), (R * T)^2, (S * T)^2, R^-5 ] = “Relators” In 2006, Marston Conder of the New Zealand Institute of Mathematics and its Applications published a list of all possible regular maps up to genus 101. He found these by an extensive brute-force computer search using concepts of group theory, and symmetry. Here is the beginning of one of several very long lists. He found a total of 6104 orientation preserving regular maps from genus 2 to genus 101. Three text lines describe every map; that is what you get. NO pictures! By the time he published this list, there were pictures for less than a dozen of these maps. -- Highlighted in yellow and magenta are so called RELATORS. They describe compound operations that bring you back to the same place in the network. -- More on these later …

Entry for the Macbeath surface of genus 7: ISAMA 2004 Macbeath Surface Entry for the Macbeath surface of genus 7: R7.1 : Type {3,7}_18 Order 1008 mV = 1 mF = 1 Defining relations for automorphism group: [ T^2, R^-3, (R * S)^2, (R * T)^2, (S * T)^2, S^-7, S^-2 * R * S^-3 * R * S^-2 * R^-1 * S^2 * R^-1 * S^2 * R^-1 * S^-2 * R * S^-1 ] This is the 2nd-simplest surface for which the Hurwitz-limit of 84*(genus-1) can be achieved. For my 2006 Bridges talk I wanted to make a nice sculptural model of this surface … I simply could not find a solution ! Even though I tried really hard … In particular, there is also an entry for the Macbeath surface of genus 7 for which I wanted to make a model since 2005. This surface is special in the same way that the Klein surface is special: it exhibits the maximum number of symmetries possible for its genus, i.e., 84(genus-1). = 504 -- nott counting reflexions. That is why I was tempted to make a sculptural model for this surface, just as Helaman did for the Klein surface -- but did not succeed. However, Conder’s list did not tell me anything that I did not already know – and there were no pictures and no hints how to make an actual embedding in a genus7 surface …

ISAMA 2004 R7.1_{3,7}_18 Paper Models Here are two of the paper models that I built in 2006. These are based on a cylinder with 7 helical handles attached. But the lines do not have the right connectivity!

Styrofoam Model for R7.1_{3,7}_18 ISAMA 2004 Styrofoam Model for R7.1_{3,7}_18 Here is one of my foam-rubber models based on a disk with 7 holes. But it also does NOT contain a proper regular map.

Globally Regular Tiling of Genus 4 On and off since 2006 I have dabbled in finding other highly symmetrical embeddings. At the Bridges conference in 2007, I presented an embedding for this globally regular tiling of genus 4. I only later found out that this is Conder’s map R4.2-dual. It is composed of 24 pentagons. As shown here on the right, it is NOT an embedding. The 12 ordinary pentagonal faces heavily intersect one another. Actual cardboard model (Thanks to David Richter) Conder: R4.2d_{5,4}_6 But this is not an embedding! Faces intersect heavily!

It works !!! A Solution for R4.2_{4,5}_6 ISAMA 2004 A Solution for R4.2_{4,5}_6 Inspiration R3.1: Petrie polygons zig-zag around arms.  R4.2: Let Petrie polygons zig-zag around tunnel walls. After several false starts, I finally found a solution based on a truncated octahedron with 4 tunnels connecting opposite hexagons. The key breakthrough came when I let the 6-step Petrie polygons zig-zag around the tunnel walls as shown on the left. A look into a tunnel It works !!!

Nice Color Pattern for R4.2_{4,5}_6 ISAMA 2004 Nice Color Pattern for R4.2_{4,5}_6 Use 5 colors Every color is at every vertex Every quad is surrounded by the other 4 colors Here is a nicely colored version. Of course the 4 tunnels would intersect at the center, if they are kept straight, And they would break the geometrical symmetry, if they were bent to avoid those intersections. We will come back to this issue later when we talk about the best visualization models.

A Graph-Embedding Problem Find surface of lowest genus in which Dyck’s graph can be drawn crossing-free Dyck’s graph = K4,4,4 Tripartite graph Nodes of the same color are not connected. And at ISAMA 2004 in Chicago I presented a solution to a graph embedding problem, which at that time I did not realize also was a solution for a regular map. It is known as Dyck’s graph. It is the tripartite graph K444, where nodes of the same color do NOT have any edges between them.

ISAMA 2004 An Intuitive Approach Start with highest-symmetry genus-3 surface: “Tetrus” Place 12 points so that the missing edges do not break symmetry:  Inside and outside on each tetra-arm. Do not connect the nodes that lie on the same symmetry axis (same color) (or this one). The way I solved this was by starting with a surface of genus 3 with the highest possible symmetry; this would be a “tetrus”, i.e., a nicely rounded tertrahedral frame. I placed 12 points with highest symmetry : 2 per arm, one inside and one oln the outside. Then I connected all points that would NOT lie on the same symmetry axis. Detailed story in the Isama 2004 proceedings or my web page.

A Tangible Physical Model 3D-Print, hand-painted to enhance colors Then I connected these points with nicely rounded edges, and colored in the triangular facets defined by them. This forms the regular map: R3.2 I made a computer model, and from that a 3D print model on which I over-paint the main features by hand. R3.2_{3,8}_6

A Virtual Genus-3 Tiffany Lamp I was talking about visualization models becoming ART. Here is one attempt at it. This is still the same model as before -- but now rendered as a glassy, transparent surface. I also added 4 light-bulbs in the tetrahedral corners. This turns it into a Tiffany Lamp!

Light Cast by Genus-3 “Tiffany Lamp” And we can go a step further … and turn on those light bulbs and simulate what the projected pattern on the walls would be. You find this picture in the art gallery. Rendered with “Radiance” Ray-Tracer (12 hours)

“Low-Hanging Fruit” Some early successes . . . ISAMA 2004 “Low-Hanging Fruit” Some early successes . . . By the Fall of 2008 I had become aware of Conder’s list and noted that there were quite a few of these regular map puzzles to be solved. There are 10 maps for genus 2, 20 each for genus 3 and genus 4 each, 26 for genus 5, and the numbers keep growing with the genus. Some of the maps were easy to figure out: like R2,2 and R3.6 shown here. They imply a pattern that tells us how to do higher genus embeddings such as R4.4_{4,10}_20 with five arms and R5.7{4,12}_12 with six arms -- and indeed those constructions have the properties specified by Conder’s list. But other entries right next in the list stubbornly refused my attempts at finding a nice embedding. R2.2_{4,6}_12 R3.6_{4,8}_8 R4.4_{4,10}_20 and R5.7_{4,12}_12

Genus 5 336 Butterflies Only locally regular ! At that time, I was particularly interested in the genus 5 cases. Since in 2007 I had been playing with LOCALLY regular tilings on cube frames of genus 5: Here is an object inspired by Doug Dunham’s “168 Butterflies on a Polyhedron of Genus 3” from Bridges 2002. Each triangle-face contains 3 butterflies. But this is only a locally regular map.

Globally Regular Maps on Genus 5 ISAMA 2004 Globally Regular Maps on Genus 5 With a lot of trial and error I later found these mappings, which I then exhibited in Banff in the art exhibit in 2009. But I also had many false starts in this – more on this later …

Emergence of a Productive Approach ISAMA 2004 Emergence of a Productive Approach Depict map domain on the Poincaré disk; establish complete, explicit connectivity graph. Look for likely symmetries and pick a compatible handle-body. Place vertex “stars” in symmetrical locations. Try to complete all edge-interconnections without intersections, creating genus-0 faces. Clean-up and beautify the model. { Look how best to turn this into “Art.” } Gradually during 2009 I developed a somewhat systematic approach that often seemed to yield success. Here is an outline: LIST… In the following I will elaborate on these steps and illustrate with examples. Last one is an optional step.

Depiction on Poincare Disk First step is to produce a Poincare disk with the right tessellation parameters. This is easy, there are many applets on the web Use Schläfli symbol  create Poincaré disk.

Relators Identify Repeated Locations ISAMA 2004 Relators Identify Repeated Locations R3.4_{4,6}_6 Relator: R s s R s s Now we need to cut out the exact domain of the map. For that we need to know which vertices and facets in the Poincare disk are repeated instances of the same vertices and facets of the map. To figure that out we need to use the "Relators" that characterize the particular map. They are given as sequences R,S,r,s ... that represent an identity transform on the map. If we apply such a string of Relator operations on the Poincare disk, starting from any point, then the landing site is another instance of that same point. One may need to apply more then one relator to cover all the area of the Poincare disk that is visible with enough resolution. It is a good idea to also run a few tests with a different relator after one is done to check that on has made no errors. Operations: R = 1-”click” ccw-rotation around face center; r = cw-rotation. S = 1-”click” ccw-rotation around a vertex; s = cw-rotation.

Complete Connectivity Information ISAMA 2004 Complete Connectivity Information Triangles of the same color represent the same face. Introduce unique labels for all edges. This should in principle allow us to color the whole disk with unique colors and labels for all unique facets and vertices. Practical difficulties: disk resolution is finite, facets get very small very quickly as we approach the rim; and there might not be enough easily distinguishable colors to color all facets on the map differently. This results in an explicit connectivity graph of the whole regular map, And it unambiguously describes the “stars” around every single vertex. It also gives an idea of what kinds of symmetries we might expect: e.g. in this case there seems to be a clear 4-fold symmetry around every vertex.

Low-Genus Handle-Bodies ISAMA 2004 Low-Genus Handle-Bodies With this we can try to pick a good handle body … There is no shortage of nice symmetrical handle-bodies of low genus. This is a collage I did many years ago for an art exhibit. There is no shortage of nice symmetrical handle-bodies of low genus. This is a collage I did many years ago for an art exhibit.

Numerology, Intuition, … ISAMA 2004 Numerology, Intuition, … Example: R5.10_{6,6}_4 Second try: tetrahedral symmetry Now comes the least-well defined part, that involves the most intuition or plain guess-work: Where to place those vertices ? If we consider R5.10 which has 8 vertices of valence 6 and also 8 hexagons as its faces, then it is rather straight-forward to pick a cube frame and place the 8 vertices at the corner. If we have a good embedding, then that same arrangement should also work for the dual configuration; thus it is a good idea to also consider where the face centers might be going. For this example, they might readily go on the inside of the 8 corners of the cube frame. Then we try to connect the vertices with edges to form twisted hexagons in such a way that ideally we obtain always the same pattern on each of the 12 struts connection the 8 corners with their neighbors. The regular pattern shown above forms! BUT it is not the right thing – When starting to check the length of the Petrie polygons, we realize that some are of length 6. This shows we did not realize the above map. -- The it occurred to me, that perhaps half the vertices should go on the inside of the corner in a tetrahedral pattern, and correspondingly, 4 face centers switch to the outside. (Fig. on right). Overall we now have tetrahedral symmetry rather than that of an oriented cube! – but now I discosvered that there were some Petrie polygons of length 2 – still no cigar ! First try: oriented cube symmetry

An Valid Solution for R5.10_{6,6}_4 ISAMA 2004 An Valid Solution for R5.10_{6,6}_4 Actually, the symmetry of the figure had to be reduced even further to that of the oriented tetrahedron, and the hexagons had to be stretched and twisted even more in order to obtain a proper solution for this map R5.10. Now all the Petrie paths are of the proper length of 4 edges. On the left you see a virtual rendering with all eight faces shown in different colors; at right is one of my paper models that was exhibited at the last Bridges conference. By now you should have gotten and idea of the main steps involved in my intuitive approach to finding such symmetrical embeddings. Virtual model Paper model (oriented tetrahedron) (easier to trace a Petrie polygon)

2 Methods to Find Embeddings ISAMA 2004 2 Methods to Find Embeddings A general “text-book” method for embedding a network in a handle-body of appropriate genus. But this will not yield any nice regular solutions! The computer-search by Jack J. van Wijk, which found more than 50 good embeddings. But not clear which solutions will emerge; some simple cases could not be found! Some solutions are more twisted than they need be. Now is the time to mention two quite different algorithmic approaches for finding embeddings of such maps on two-manifolds. Unfortunately, 25 minutes is not enough to explain these methods so that you will understand them. Please read the paper to get a rough idea, and if you are still interested, read the cited references. Here I just give you a quick idea what has been done…

The General Text-book Method (1) ISAMA 2004 The General Text-book Method (1) Convert the domain of the regular map to a special 4g-gon with the edge sequence: ( a, b, a’, b’ ) g Maintain all copies of the vertex “E” on the perimeter, but get rid of all the other vertices, by burying them inside the polygon, by repeatedly cutting off some portion like the purple area around D on the right and re-gluing it on the other side to the equivalent edge segments.

The General Text-book Method (2) ISAMA 2004 The General Text-book Method (2) In the general case g handles will result. Now each sequence of 4 edges: ( a, b, a’, b’ ) is first closed into a tube by joining a to a’, then into a closed handle by joining b to b’.

The General Text-book Method (3) ISAMA 2004 The General Text-book Method (3) Here is an explicit genus-2 net with this kind of perimeter, and the folded-up 2-loop handle-body that results. This shows the folding phase for a nice polyhedral genus-2 realization.

The General Text-book Method (4) ISAMA 2004 The General Text-book Method (4) Unfortunately, the result is not nice ! Result for R2.1_{3.8}_12 Vertices do not end up in symmetrical places. The local edge-density is quite non-uniform. Unfortunately, while this method is general, it does not give us what we really want. This is the result of phase 1 for the map R2.1_{3,8}_12. As you can see: the vertices … Edge-density …

Jack J. van Wijk’s Method (1) Starts from simple regular handle-bodies, e.g. torus, “fleshed-out” hosohedron, or a Platonic solid. Put regular edge-pattern on each connector arm: Determine the resulting edge connectivity, and check whether this appears in Conder’s list. If it does, mark it as a success! Here is a computer-based method by Jack van Wijk. He starts from simple regular handle bodies… And places regular edge patterns on each connector arm. Then he determines the resulting edge connectivity and checks whether this appears in Conder’s list.

Jack J. van Wijk’s Method (3) Cool results: Derived from … This results in some beautiful virtual models. On the left is one based on a dodecahedral frame. On the right is one based on a torus with a 3 by 3 square tile array. Dodecahedron 3×3 square tiles on torus

Jack J. van Wijk’s Method (2) ISAMA 2004 Jack J. van Wijk’s Method (2) For any such regular edge-configuration found, a wire-frame can be fleshed out, and the resulting handle-body can be subjected to the same treatment. It is a recursive approach that may yield an unlimited number of results; but you cannot predict which ones you will find and which will be missing. You cannot (currently) direct that system to give you a solution for a particular map of interest. The program has some sophisticated geometrical procedures to produce nice graphical output. The clever thing is that his method is recursive … Here are some sample results …

J. van Wijk’s Method (5) Cool results: Embedding of genus 29 The highest genus result that he presented: Cool results: Embedding of genus 29

Jack J. van Wijk’s Method (5) Alltogether so far, Jack has found more than 50 symmetrical embeddings. But some simple maps have eluded this program, e.g. R2.4, R3.3, and: the Macbeath surface R7.1 ! Also, in some cases, the results don’t look as good as they could . . . In summary: …

Jack J. van Wijk’s Method (6) ISAMA 2004 Jack J. van Wijk’s Method (6) Not so cool results: too much warping. For the map R3.8: On the left, Jack’s result with an unnecessary amount of twisting. On the right is my much more regular solution on a tetrahedral frame. My solution on a Tetrus

Jack J. van Wijk’s Method (7) ISAMA 2004 Jack J. van Wijk’s Method (7) Not so cool results: too much warping. Now for the map R3.11: Again, on the left, Jack’s result with too much twisting. On the right is my own solution based on one of the several generic patterns that I found during the last 18 months … So it is appropriate to look at one of these generic patterns which I call “vertex flower”. “Vertex Flower” solution

“Vertex Flowers” for Any Genus ISAMA 2004 “Vertex Flowers” for Any Genus This classical pattern is appropriate for the 2nd-last entry in every genus group. All of these maps have exactly two vertices and two faces bounded by 2(g+1) edges. … even for genus = 1 These structures are self-dual, so it is natural that one vertex lies always inside the pole, and the other one on the outside, and opposite each pole-vertex is one of the face centers. g = 1 g = 2 g = 3 g = 4 g = 5

Paper Models for “Vertex Flowers” ISAMA 2004 Paper Models for “Vertex Flowers” g = 2 g = 3 But these computer models were not the first thing I constructed. First I made some topological models out of paper strips, As shown here for the genus 2 and 3 cases. Let’s take a closer look at the construction of the genus-2 model I first found those embeddings with these paper strip models.

Anatomy of a Paper-Strip Model ISAMA 2004 Anatomy of a Paper-Strip Model bend and glue Here is how I built the topological model for the genus-2 case. I knew it consisted of two hexagons, connected alternatingly to two vertices. So I made these spidery hexagons and fit them together. I checked the Petrie polygon length on this structure, and only then created the computer model. -- This was an easy example. Now let me show you how I struggled with a more difficult one … R2.5_{6,6}_2

ISAMA 2004 The Regular Map R3.3_{3,12}_8 16 triangles, 24 edges, 4 vertices (valence-12). This is the more difficult problem: Here you see the result of the first phase: establishing explicitly the complete connectivity for R3.3.

Deforming & Folding the Map Domain Here is the cutout of the whole net -- deformed to fit into a rectangle that can be folded up into a torus with two openings. Then I forced the 4 green vertices to come together… Fold into a torus with two openings: connect horizontal and vertical edges of same color; Then bring the 4 green vertices together . . .

Evolution of the Topology Model ISAMA 2004 Evolution of the Topology Model In the middle you see a roughly toroidal shape. Two of the slits in it appear at the top and bottom near the center. They carry instances of the same map edges. To merge them, I needed to add two more handles to the basic torus – making it now a genus-3 object. The green vertex appears at the edges of all those openings. The two bottom one and the two top ones can readily be merged. But then to merge the top and bottom instances, one pair has to be pulled along the two new handles, thus greatly stetching and deforming some of the attached edges. But when done, it clearly tells me what edges have to run along which handles, and in which sequence they appear on these handles. Overall we now have a handle body with 4 handles and two 4-way junctions, with two vertices ate each of the two junctions. On the right, you see a new cleaned-up model that brings out those topological properties.

Models for R3.3_{3,12}_8 Net and paper model ISAMA 2004 Models for R3.3_{3,12}_8 Based on that information, I could now draw a clean net for a genus-3 body based on the 4-arm hosorus, and from that build a small paper model that exhibits maximal symmetry for this configuration. Net and paper model

ISAMA 2004 Models for R3.3_{3,12}_8 Tried other variations by drawing on plastic tubing. Study symmetry breaking … Alternative model in which the four vertices have been moved to the middle of the handles. Original clean paper model

Good Solutions Can Be Re-used ! A key insight was that some solutions readily lead to other ones. Here I merged neighboring triangles into quads, by eliminating some edges. The maps R3.3_{3,12}_8 and R3.5_{4,8}_8 are related. Pairs of triangles turn into quadrilaterals.

Re-use of R3.3 Topology for R3.5 ISAMA 2004 Re-use of R3.3 Topology for R3.5 This is the result for another regular map. Re-use is important. A good “arsenal” of solutions makes it easier to find other solutions. These “diagonal” edges are no longer present

Visualizing R4.2_{4,5}_6 A transformation maintaining symmetry ISAMA 2004 Visualizing R4.2_{4,5}_6 Now a comment about visualization: Of course the 4 tunnels would intersect at the center, if they are kept straight, and it would break the geometrical symmetry, if they were bent to avoid those intersections. So let’s cut the tunnels in the middle and pull individual half-tunnels outward and joint opposite stubs into handles. Topologically that is still the same object. But bending these handles one way or another will violate the octahedral symmetry. The only way we can maintain symmetry if we keep the handles straight, but make them infinitely long and close the handles through infinity. OR … A transformation maintaining symmetry

ISAMA 2004 R4.2_{4,5}_6 Lattice Or . . . If we connect identical atoms into a regular lattice. This allows us to see all the connections – and we don’t have to reduce the symmetry of our display. I was inspired to the possibility to display these regular maps on regular lattices by Dirk Huylebrouck’s paper on regular polyhedral lattices later in this conference. I think there is a whole set of interesting research to examine those Platonic or Archimedean lattices, which clearly have local regularity, also for their global regularity.

R4.2_{4,5}_6 Lattice (wide angle) ISAMA 2004 R4.2_{4,5}_6 Lattice (wide angle) This kind of lattice display also has another advantage: By using perspective and a wide-angle view, ore than 50% of the atom surface can be inspected. With the proper parameter settings on the lattice and on the camera, eye-catching displays can be generated. I think a door to a whole other world has just been opened . . .

ISAMA 2004 More … I have much more to say on the subject of embedding regular maps. It was difficult to choose only about 50 pictures from about a thousand pictures of Poincare disks, physical models and virtual renderings – and construct a cohesive 25-minute story. I will definitely report on these matters again in the future.

Conclusions “Doing math” is not just writing formulas! ISAMA 2004 Conclusions “Doing math” is not just writing formulas! It may involve paper, wires, styrofoam, glue… Sometimes, tangible beauty may result ! For the people in the audience who may have been scared to go into mathematics because it is supposedly so rigid and formal, this talk may have given you a different view. Often the most creative and fun part of mathematics is very unstructured and full of unconventional steps. Paper strips, wires, balls of clay or pieces of styro-foam may be the most effective tools in this early exploratory phase. Then, finding a neat solution for a tough problem has a sense of beauty that is difficult to describe to someone who has not played a part in that process. But hopefully some of the resulting artifacts can stand on their own and are enjoyable by themselves. This is something I plan to work on in the near future.