1. Modular Toroids Constructed from Nonahedra
Florida 1999
Bridges, Stockholm, 2018
Modular Toroids Constructed from Nonahedra
Our work on modular toroids has been inspired by a piece of art work at the 2016 Bridges art exhibition.
Yifat Amir & Carlo H. Séquin
EECS Computer Science Division
University of California, Berkeley

2. Bente Simonsen: “Torus” (2016)
Granada 2003
Bente Simonsen: “Torus” (2016)
In 2016, Bente Simonsen presented this sculpture, called “Torus”. It was explained as follows:
“The sculpture is created from five nine-sided polygons, each with 3 square sides and 6 pentangle sides, all with the same side length, and fitted together to build a pentagonal torus.”
“The sculpture is created from five nine-sided polygons, each with 3 square sides and 6 pentangle sides, all with the same side length, and fitted together to build a pentagonal torus.”

3. The Nonahedral Building Block
Granada 2003
The Nonahedral Building Block
If all faces were indeed regular n-gons, then this would be a Johnson solid.
Let’s take a closer look at the 9-sided building block consisting of 3 squares and 6 pentagonal faces. If all faces were indeed regular n-gons, then this would be a Johnson solid.
>>> But it does not work out that way. If we put 3 regular pentagons together to form a cap (as in the Platonic dodecahedron), and then put two such caps together, then the resulting openings on the sides are not square; >> they are not even planar!
>>> Or, if we start with 3 perfect, vertical unit-squares, placed at angles of 120 against one another, and separated by one unit edge-length between their corners; -- and then further add the top and bottom pyramid vertices also with unit-length edges,
>>> then the resulting 5-sided faces are also not planar. -- In summary, the described building block does not exist! -- It is a Near-miss Johnson solid.
But the described object does not exist! It is a Near-miss Johnson solid.

4. Revisiting the “Torus”
Granada 2003
Revisiting the “Torus”
60
Something does not quite add up …
D3-symmetry of building block implies:  60 between square faces. This would then lead to a 6-module ring! --
Let’s take another look also at the complete “Torus” sculpture: The 3-fold D3-symmetry of the building block implies that there are angles of 60 between the square contact faces. This would then naturally lead to a 6-module ring!
So there is some “cheating” going on! -- How come, we do not notice these geometrical imperfections --??

5. Hiding the Flaws in the Module
Granada 2003
Hiding the Flaws in the Module
Visually optimized
Many different flaws:
Irregular pentagons
Irregular square faces
Square faces at bad angles
Overall lack of 3-fold symmetry . . .
There could be many different flaws in these models: -- Irregular pentagons; -- Irregular square faces; -- Square faces at bad angles; -- Overall lack of 3-fold symmetry . . .
Perhaps, if we deviate in all these characteristics by just a little bit, none of these flaws becomes too obvious, and we still get an overall visibly-pleasing result. When one first looks at Bente’s Torus, one does not realize that there are any imperfections. So, how did she do this ??
-- How can we find the “most acceptable” solution?

6. Parameterized Module Description
Granada 2003
Parameterized Module Description
The nonahedral module has 14 vertices. Each vertex needs 3 coordinate values; for a total of 42 numerical parameters!
Exploiting 12-fold D3h-symmetry leaves us with only 4 parameters. This is a manageable optimization space! Paper describes one way of assigning parameters:
ex = distance of squares from origin
rc = radial distance of corner
hc = height of corner
tz = height of pyramid tip
First we need a Parameterized Module Description, so that we can change some numerical values and examine what effect this has on the visual quality of the resulting module. The nonahedral module has 14 vertices and each one has 3 coordinate values, --for a total of 42 numerical parameters! But it would be insane to try to jiggle all 42 parameter values. Enforcing 12-fold symmetry for this object, reduces the number of independent parameters to only FOUR! This now is a manageable optimization space. These parameters can be introduced in different ways; the paper describes one possible way of assigning them; this is illustrated here . . .

7. The Penalty (or “Cost”) Function
Granada 2003
The Penalty (or “Cost”) Function
Trying to minimize all the possible flaws listed in the previous slide: we combine them into one overall penalty function:
ELD = Edge-Length Differences:
avl = ∑ (edge-lengthi) / 21; ELD = ∑ (lengthi – avl)2
PAD = Polygonal Angle Deviations:
PAD = ∑ (angles4 – 90)2 + ∑ (anglep5 – 108)2
FNP = Face Non-Planarity:
FNP = ∑ (disti)2 {distances of vertices from best-fitting plane}
COST = α * ELD + β * PAD + γ * FNP
Now, by varying these four parameters, we try to minimize all the possible flaws listed in an earlier slide. We will do this by combining all of them into one overall penalty- or cost- function. For instance, we can define an overall “COST” that measures the visual deviation from perfection by forming a weighted sum of three different terms:
First there are the Edge-Length Differences (ELD); second there are the Polygonal Angle Deviations (PAD); and finally, there are the Face Non-Planarity (FNP). The three coefficients α, β, and γ are left to be set by the artist.

8. The Penalty (or “Cost”) Function
Granada 2003
The Penalty (or “Cost”) Function
Visible flaws = “Cost”
Param. #1
Param. #2
Cost function is a “hilly landscape” over our 4-dimensional parameter space. (In the example above we only show 2 dimensions)
This defines the COST for all possible combinations of parameter values. Graphically, this forms a “hilly landscape” in our 4-dimensional parameter space. -- But in the example here, we only show only a 2-dimensional parameter space.
The goal now is to find the lowest point in this landscape.
>>> One way to find this, is to perform optimization by gradient descent. Conceptually, you can think of this as starting with a reasonable sampling point, corresponding to a combination of parameters that are not too far off from an acceptable solution.
>>> There we insert a drop of water and watch it run down-hill to the lowest point in the neighborhood.
In “landscape” do gradient descent optimization (= follow a drop of water to the lowest point it can go to)

9. Optimizing a Single Nonahedron
Granada 2003
Optimizing a Single Nonahedron
However, it is not clear, how the individual geometrical flaws should be combined into a single cost (or penalty) function. Different values for alpha, beta, and gamma change how much each flaw impacts the value of the overall cost. These values can be set to prioritize different visual flaws. For example, if this object is made with a shiny, reflecting surface, then face-planarity may be very important, since non-planar faces would distort reflected lines.
Different participants in our study group tried different cost functions with different weights for the individual flaws. Prof. Sequin also tried to simply fiddle manually with 4 sliders defining the numerical values of the 4 parameters, while observing the displayed nonahedron. With only 4 parameters, this is a quite practical approach.
Different optimization results (sorted by z-height).

10. Constructing an Optimal(?) “Torus”
Granada 2003
Constructing an Optimal(?) “Torus”
Follow the artist: Keep all squares planar and regular.
Place 5 contact faces with circular 5-fold symmetry.
Adjust distance to outer squares.
Position top and bottom pyramid vertices (2 parameters).
Now we take a similar optimization approach for the construction of a pentagonal torus as a whole. Here the basic nonahedral building block must be further distorted: To construct a ring with nice 5-fold symmetry, we need to put the contact faces at angles of 72 degrees against one another.
In a first approach, we followed the artist, and have kept all squares planar and regular. – We place the 5 (green) contact faces with circular 5-fold symmetry around the white pentagonal hole. -- Then we introduce one parameter to adjust the distance to the outer (red) squares.
>>> Finally, we place the top and bottom pyramid vertices; this introduces another two parameters. Here we trade off constancy of edge lengths against the planarity of the pentagons.
>>> The right-most figure shows a solution where the pyramid vertices were adjusted so as to keep all faces planar, -- at the cost of longer edges leading to the pyramid tips.

11. Granada 2003
Another Optimization
The 5 quadrilateral contact faces are not directly visible, so it is hard to judge, whether they are perfectly square. They could be rhombic! This yields an extra DoF.
The 5 quadrilateral contact faces are not directly visible; thus it is hard to judge, whether they are perfectly square. They could be rhombic! This yields an extra degree of freedom (DoF). This allows us to obtain planar faces AND a smaller deviation in edge length. Here is a corresponding 3D print.
>>> Once the math for this type of optimization is in place, the same approach can also be used to construct a 7-module torus. We just need to start with 7 rhombic faces in a circle around a heptagonal hole.
>>> Here again is a solution where all faces were kept planar.
Once the math for this type of optimization is in place, the same approach readily yields a 7-module torus. We just need to start with 7 rhombic faces in a circle.

12. Play with the Nonahedron Module!
Granada 2003
Play with the Nonahedron Module!
Can we make an 8-module ring?
Now that you understand, how we create these various models and make them look as good as we possible can, lets go wild and play with this basic nonahedron module in order to create other tori -- and also objects of higher genus!
A first question: Can we also make a reasonably good looking torus composed of 8 modules?
YES we can! But now it is better to use the pentagons as contact faces! -- This forces smaller distortions onto the individual modules. In the left figure you can see that 8 of the individually optimized nonahedron modules fit together almost perfectly.
-- To construct the complete toroid, we start by placing the 8 pentagonal (green) contact faces around an octagonal hole. These green pentagons just need to be planar, not regular! Their exact dimensions are optimized to make the visible faces look as good as possible.
[Green pentagons use 3 params. The tip of the yellow pentagons, needs 2 params. To define the outermost blue edges needs another 3 parameters].
YES! But it is better to use the pentagons as contact faces!
This forces smaller distortions onto the individual modules.
The green pentagons just need to be planar, not regular!

13. Higher-Genus Structures
Granada 2003
Higher-Genus Structures
Now we want to use the nonahedron module to compose structures of higher genus – this means donuts with multiple holes. As a first step, we just use the 3-fold symmetrical module to construct hexagonal toroids, and then combine them into arbitrarily complex planar networks. – No adjustments are needed; the nonahedron module has just the right kind of symmetry!
A first step: Planar hexagonal networks.
Readily use the individually optimized module.

14. Granada 2003
3D Cage Structures
Vertical connections based on another 8-module toroid.
From here it is just a small step to expand these networks into the 3rd dimension and to form 3D Cage Structures.
To make a connector branch, that goes off in the 3rd dimension, we use the pentagonal faces, which are already close to 45 degrees with respect to the horizontal symmetry plane. A connector of two such modules (between the two cyan pyramid tips) is part of yet another 8-module toroid, in which the contact faces alternate between squares and pentagons, as shown in the centre. On the left is a tight genus-2 structure where the 3 connectors are attached directly to the cyan top and bottom modules. On the right is a looser structure, where the vertical connectors are formed by half of an 8-module toroid.
Contact faces: square pentagonal

15. 3D Cage Structures (2) A denser configuration …
Granada 2003
3D Cage Structures (2)
Of course, we can combine both types of vertical connectors and obtain a structure of genus 5.
And, in principle, we can take two copies of an arbitrarily complex planar network formed with hexagonal rings, and put several of these vertical connectors between them, -- making structures of arbitrarily high genus.
And, we can also start with hexagonal networks in multiple planes and expand this structure in the vertical direction.
A denser configuration …
Expanding the network …

16. Structures Based on Platonic Solids
Granada 2003
Structures Based on Platonic Solids
Now let’s take a different approach and see what we can do, if we are willing to distort our nonahedral modules more seriously, in order to obtain structures of higher genus that also display a maximal degree of symmetry. Here it is advantageous to start with the symmetries of the Platonic solids. --- For instance, with a cube. Here are two solutions: 1.) The convex hull is kept to be a cube; 2.) it is a more rounded object.
On the right of the two main figures, you can see the shape of the individual nonahedral modules. The rounded cube frame uses a more distorted module. But, clearly, both of them are now quite far from being a near-miss Johnson solid!
Cube and cuboid genus-5 structures with square and rhombic contact faces, respectively.

17. Structures Based on Platonic Solids
Granada 2003
Structures Based on Platonic Solids
As a second example: we can create a highly symmetrical genus-11 structure by placing a nonahedral module at each vertex of a pentagonal dodecahedron. The necessary module is a (affinely?) distorted version of the original near-miss Johnson solid.
The right-hand model was printed as two halves of the whole thing. (This made it easier to remove the supportive scaffolding.)
Dodecahedral genus-11 structure composed from 20 nonahedral units.

18. Structures Based on Platonic Solids
Granada 2003
Structures Based on Platonic Solids
A third example can be constructed in a similar fashion, starting from a tetrahedron. The resulting module is shown to the right of the modular, genus-3 tetrahedral structure.
Two views of the tetrahedral frame One module.

19. Granada 2003
Summary
We have investigated the challenges and trade-offs in making many “miracle” sculptures composed of a nine-sided, near-miss Johnson solid module.
Each sculpture yields a different set of constraints which give rise to a new set of parameters and a newly visually “optimized” module.
Hopefully you have gained insight into how we unraveled the mysteries of Bente Simonsen’s original sculpture!
We have investigated the challenges and trade-offs in making many “miracle” sculptures composed of a nine-sided, near-miss Johnson solid module.
Each sculpture yields a different set of constraints which give rise to a new set of parameters and a newly visually “optimized” module.
Hopefully you have gained insight into how we unraveled the mysteries of Bente Simonsen’s intriguing original sculpture!

20. Granada 2003
Acknowledgements
We are grateful to Bente Simonsen for displaying her “Torus” at the 2016 Bridges Art Exhibit, which lead us onto this path of exploration.
My co-workers: Ruta Jawale, Hong Jeon, Alex Romano, Rohan Taori, and our adviser Prof. Sequin.
More artwork displaying “Geometric Impossibilities” can be found at Simonsen’s home page:
We are grateful to Bente Simonsen for displaying her “Torus” at the 2016 Bridges Art Exhibit, which lead us onto this path of exploration.
And I would like to acknowledge the work of my co-workers and of our advisor.
More intriguing artwork, displaying “Geometric Impossibilities” can be found at Bente Simonsen’s home page: