1. Bridges Baltimore, July 2015
Florida 1999
Bridges Baltimore, July 2015
Large, “7-Around” Hyperbolic Disks
Good afternoon. I am Sean Liu, and I am going to tell you about a project that I did with two fellow students, Young Kim and Raymond Shiau, at UC Berkeley and our mentor, professor Carlo Sequin. We are trying to make a discrete computer model of the hyperbolic plane, composed of as many equilateral triangles as possible. Here you see 3 hyperbolic models made by other people – and I will discuss those shortly.
Sean Jeng Liu, Young Kim,
Raymond Shiau, Carlo H. Séquin
University of California, Berkeley

2. Assembling Equilateral Triangles
Granada 2003
Assembling Equilateral Triangles
5 per vertex: pos. curved:  Icosahedron
6 per vertex: flat:  a plane
7 per vertex: neg. curved:  hyperbolic
But first a brief review of what we mean when we try to construct a hyperbolic disc from all equilateral triangles.
When we make a regular assembly of such triangles where we place always FIVE triangles around a vertex, we obtain a surface with positive curvature: the icosahedron; this is one of the Platonic solids.
When we consistently group SIX triangles around each vertex, we obtain a regular flat tiling.
When we try to place SEVEN equilateral triangles around a vertex, we obtain something that looks like an asymmetric, warped “potato chip,” forming a negatively curved saddle surface. Nevertheless, we can continue the process and place seven triangles around every vertex along the perimeter. The resulting geometry gets rather convoluted rather quickly!

3. Hyperbolic Surface: Poincaré Disk Model
Granada 2003
Hyperbolic Surface: Poincaré Disk Model
But the topological structure is totally regular and can be captured rather nicely in the Poincare disk, if we scale down the triangles as we get further away from the center.
This scaling avoids over-crowding near the perimeter. With every annulus of triangles that we add around the central seven, the diameter of the disk grows linearly, but the number of triangles in each ring grows exponentially. So, eventually, the process will suffocate, if the triangles are all the same size.
In this project it is exactly our goal to see how far we can push this growth of the disk without causing any self-intersections among the equilateral triangles.
Scaling allows to accommodate infinitely many triangles.

4. Granada 2003
How much of that infinite tiling can we accommodate with equilateral triangles ?
Here are two results by people who have done this experiment in a physical manner using small paper triangles.
These results have been found on the web when entering “hyperbolic surface”.
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Hyperbolic surface constructed by Amy Ione and CW Tyler (2001)
David Richter (see ref in paper)
Amy Ione & CW Tyler David Richter

5. Better Luck with Soft Materials
Granada 2003
Better Luck with Soft Materials
Here is one of many attempts to model a portion of the infinite hyperbolic plane by using crocheting. Using a soft, pliable material like wool allows one to push the congestion-limit further out, since adjacent lobes can interactively be pushed out of the way where needed.
On the right is a computer simulation of such a smooth approximation of a hyperbolic disk found on the web << >>
Unfortunately, no information is available as to how and by whom it has been generated.
Gabriele Meyer (posted by Loren Serfass)

6. Computer Model by Mark Howison (2007)
Granada 2003
Computer Model by Mark Howison (2007)
However, in this paper we really want to focus on approximating the hyperbolic disk with rigid equilatral triangles. Mark Howison, a former graduate student at UC Berkeley took a first stab at writing a computer program that could create such a structure. His program was quite sophisticated. It used simulated annealing to explore the combinatorial space of possible angular configurations between triangles as they are added to the surface. Nevertheless, eventually the spatial crowding became so intense, that the program spent most of its time backtracking, trying to undo bad configurations that preserve no space for new triangles, and trying to establish a looser, more promising configuration. At 810 triangles, the search tree is very deep, and the number of possible branches to be explored becomes astronomical! Why can we not do better? It seems there are plenty of spaces where we could add more triangles. -- For instance, we could just run an individual strip of triangles off to infinity … Thus we need some rule to prevent such “partial instantiation” of a cohesive hyperbolic disk.
Best result: 810 Triangles (20 hrs CPU time)

7. Granada 2003
Extending the Disk . . .
By adding full annuli – one at a time, with ever more triangles …
Exploit: 6-fold D3-symmetry
We must complete one annulus, before we are allowed to start the next one! If we started with a single magenta triangle, then we must add the complete annulus of all orange triangles before we are allowed to add any yellow triangles. If we cannot finish a particular annulus, because of congestion – we have to stop, and we are done! This diagram also shows our new approach: We start with a central TRIANGLE, rather than a central VERTEX. With this set-up we can force that the overall figure has 6-fold symmetry – so called D3 symmetry. Why is this a good thing? – It may seem that by demanding symmetry, we add more constraints and reduce the freedom to grow the disk as we want … This is true –- But more importantly, we will have to perform the expensive search and construction algorithm for only one sixth of the whole disk!
In the art exhibit I saw that Robert Fathauer and Marla Peterson have collaborated to create a more structured, 3-fold symmetrical model of the hyperbolic disk.

8. New Approach: Add 6-fold Symmetry
Granada 2003
New Approach: Add 6-fold Symmetry
This new approach changes somewhat the growth rate of the disk and how many triangles we gain with each annulus.
On the left is Howison’s approach; it maxed out at 810 triangles.
The figure on the right has 6-fold symmetry; and with our new approach we have been able to place 2,197 triangles.
810
2197
Howison’s annuli: – Our new annuli: starting with a central vertex starting with a central triangle.

9. Starting with a Symmetrical Core
Granada 2003
Starting with a Symmetrical Core
D3 symmetry forces some constraints:
The 4 central triangles are coplanar!
Yellow-olive edges lie on symm. axes; adj. triangles are coplanar.
To obtain such a 6-fold symmetrical solution, we start with the first magenta triangle and place it centered at the origin. There are three C2-symmetry axes passing through its center, shown as black lines. To maintain D3 symmetry, the three red triangles surrounding the original one must also lie in the same plane.
The next 12 triangles that complete the first annulus have to be placed in such a way that the shared edges between the yellow and olive triangles also lie on the symmetry axis and that the triangle pairs that share those edges are coplanar. This leaves only two (mirror-symmetric) solutions up to this point.

10. Constructing an Extended Core
Granada 2003
Constructing an Extended Core
Blue-teal edges lie on symm. axes. The two triangles are coplanar!
Give #4 & #2 the same sign for the z-value to make “nice, looping arch”
We could now use this 16-triangle “core” as a starting base and let a random search algorithm similar to Howison’s program calculate the next annulus sector, which would have as an outer border the vertex chain from #3 through #8. However, since there are still a substantial number of constraints in this ribbon domain, we decided to also complete this swath by hand. The two triangles (blue and teal) sharing vertex #3 also share an edge on a symmetry axis and thus must be coplanar; and there are again just two possible solutions for the position of vertex #4. We choose to give vertex #4 the same sign for its z-coordinate as for vertex #2; this produces a nice tall arch in the z-direction with ample room for the next annular ring to undulate around this looping border.
This can be seen in the next slide.

11. Manually Constructed, Extended Core
Granada 2003
Manually Constructed, Extended Core
coplanar
Here is a more oblique, perspective view of the manually constructed, extended core.
We made some further arbitrary decisions in the construction of this core. For instance, we choose to make coplanar the two pairs of triangles marked with green arrows; this choice leaves a 90° open angle between them in their shared plane; this can be spanned by a nice “gabled roof” with a good dihedral angle. -- In another place, where 3 triangles had to be accommodated, we chose to construct a symmetrical 3-panel “Mansard roof.”
We always start from this fixed, symmetrical “core” when we try to construct larger hyperbolic disks.
 61 triangles with D3 symmetry with nice undulating border.

12. Step-by-Step Construction
Granada 2003
Step-by-Step Construction
Complete one vertex at a time: “3”, “4”, “5”, “6”, “7”, “8” in orange “swath #1” throughout a 60 sector.
e.g. vertex “5”:
add “a”, “b”, “c” ind.; last two as a “gable.” 5 a b c
given
Triangles are added to the disk individually or in pairs along the current open border, forming a new annular ribbon.
Along the border, one vertex at a time, we complete the fan of seven triangles around every vertex. The last two triangles in each 7-element fan have to be placed as a pair, since they need to close up seamlessly with the first triangle in the fan; they form a “gabled roof” in the remaining angular space around the current vertex, and there is only a binary choice of angling the roof upwards or downwards.

13. Interference Checking & Back-Tracking
Granada 2003
Interference Checking & Back-Tracking
Interference:
Check for intersections between triangles;
Apply conservative proximity check for inner swath;
If fails, we try different dihedral angles:
Randomly; or
Direction-specific (as determined by program); or
Manual tweaking upon visual inspection
Back-track if:
Fails intersection checks after multiple tries; or
Fails to meet other heuristic guidelines
For each triangle that we place, we must check that it does not intersect any other triangles. But just checking for current intersections is too shortsighted; near-misses will definitely cause problems in the next outer annulus. Thus we do a more conservative proximity check for the current swath, and use the more extensive (and expensive) intersection test for the outer swaths If the distance between the centers of two triangles falls below a certain threshold, then we try out a number of different dihedral angles. If they still fail after a certain number of attempts, we backtrack and reposition the previous triangle.

14. Overall Strategy: 1 Annulus at a Time
Granada 2003
Overall Strategy: 1 Annulus at a Time
We only have to construct one 60 sector of the whole disk, which then gets replicated 6 times.
We construct this one “swath” (= 1/6 annulus) at a time; we try to construct a “good” swath, one that leaves most space for subsequent one.
Such a good swath gets added to the “core”; it now acts as a starting point for the next swath.
With the above heuristics in place, we now run the program many times to complete the current swath in a 60° sector around the core’s border. We visually inspect the results, and among several possible solutions pick the one that we think will form the most beneficial border for the subsequent swath. Such a good solution can then be “frozen” and added to an extended starting core. Then we can start working on the 60* swath of next outer annulus.

15. Granada 2003
Results May July 2015
With this approach we have constructed good solutions for swaths #1 (120 Triangles, shown in green) and for swath #2 (315 Triangles, shown in blue). Eventually, we also completed swath #3 (purple), by loosening some original constraints for tweaking and backtracking. Since then we have been working on swath #4 (shown in gold), -- Filling in triangle strips from both ends, as of a week ago, we are able tp place 147 triangles in this sector. Altogether this gives us 2,197 Triangles.
<<<
Here we first show the core, then we add the individual swaths green – blue – purple. Finally we show the triangles in one sector in swath #4 in gold
>>>

16. Granada 2003
Video
This shows the current state of affairs. We wiggle the display around so that you get a better understanding through the process “Shape from motion” wired into the human brain.
First we see the 61-triangle symmetrical core that we started out with. Here are the triangles of swath #1 colored in green; first in one sector, then replicated in all six sectors. As you can see, individual triangles connect to the border with just a single edge, and thus there is a degree of freedom concerning the dihedral angle at the junction. we tried to avoid forming acute dihedral angles when placing triangles in this swath, because they may make it difficult to place triangles in that region in the subsequent annulus. (181 triangles)
Here we have swath #2 colored in blue. Again, we tried to have the triangles form a border that would leave as much room as possible for future triangles in the subsequent swath. (496 triangles). This is swath #3 in purple. Here we cared less about forming a smooth border and focused on completing the swath without intersections. (1321 triangles)
And finally we have swath #4 in gold. We placed triangles starting from both ends of the sector and tried to fit as many as possible without triangle intersections. (2197 triangles)

17. Fully Instantiated Disk (May 2015)
Granada 2003
Fully Instantiated Disk (May 2015)
This is the result we obtained in May 2015, where we had a total of 964 triangles in our hyperbolic disk. Since then, we have continued “fine-tuning” our programs, and we tried to find the best values for the parameters in our various heuristic subroutines.

18. Final Result at Conference Time
Granada 2003
Final Result at Conference Time
At the time of the conference, we were able to have 2,197 triangles in our hyperbolic disk.
<< CHANGES: The rule of having degree gap for last two triangles in a fan now only applies to swath 1 (as opposed to all swaths). If, after tilting the new triangle in steps of 20 degrees for 5 times, the triangle still fails the conservative proximity check, then backtrack (if swath 1) or perform expensive test (if not swath 1) to see if there is actual intersection.
If swath >= 2 and fails proximity check for gabled roof, instead of rejecting the roof entirely, perform the expensive intersection test to see if there was actual intersection. If not, then leave it be instead of backtracking. For swath #3, got rid of +/- 90 degree constraint for manual tweaking and disabled immediate backtracking If an intersection is detected for the randomly placed triangle. (reason is that at such a tight space, randomly placed triangles almost always have intersections, so the backtracking feature is not as effective as manual tweaking)
2197 triangles

19. Granada 2003
Conclusions
Computers are useful and powerful; but brute-force approaches may only get limited results.
Use your brain to gain an understanding of the problem, and the tailor your search to make use of such insights.
A good combination of the two approaches can then result in a more effective search, reducing computation time exponentially.
This is often true in engineering problems relying on simulated annealing or on genetic algorithms.
With this project we confirmed a generally useful insight: Computers are handy tools and have become very powerful. A lot of good results can be obtained by just brute-force searching. However, when you also bring the human mind to bare and make use of the deeper understanding of the problem obtained in this way, then you can make your search smarter and much more effective. Here, by introducing and exploiting a maximal amount of symmetry, we could reduce the search tree by a factor of six; And since the size of the search tree contributes exponentially to the required computation time, the gain was substantial.
This general approach of integrating a deeper understanding of a problem with the power of the computer is particularly useful in many engineering problems that are being solved with simulated annealing or with genetic algorithms.