Add the configuration space of equiangular equilateral hexagons

Test-problems.md

@@ -2,12 +2,12 @@ Here are some objects one might explore and problems one might solve with dyna3.
 
 ## Basic elements and constraints
 
-### Rigid Hexagon
+### Rigid hexagon
 
 #### Source
 
 - **Author:** Noam Elkies
-- **Published:** P12 from [the Playground in *Math Horizons*: Volume 2, Issue 1](https://www.tandfonline.com/doi/abs/10.1080/10724117.1994.11974897)
+- **Published:** P12 from [the Playground in *Math Horizons* 2 (1)](https://doi.org/10.1080/10724117.1994.11974897)
 
 #### Statement
 
@@ -21,6 +21,27 @@ AD &\;=\;& BE &\;=\;& CF &\;=\;& r + s
 ```
 for some positive $r$, $s$. Show that $A$, $B$, $C$, $D$, $E$, $F$ must be the vertices of an equiangular planar hexagon.”
 
+## Exploring configuration spaces
+
+### Equiangular equilateral hexagons
+
+#### Source
+
+- **Author:** Jun O'Hara
+- **Published:** ["The configuration space of equilateral and equiangular hexagons."](https://doi.org/10.18910/25093) Osaka Journal of Mathematics 50 (2)
+- **Access:** Also available from [Project Euclid](https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-50/issue-2/The-configuration-space-of-equilateral-and-equiangular-hexagons/ojm/1371833496.full)
+
+#### Statement
+
+O’Hara has described the configuration space of hexagons in 3-space which are equiangular, equilateral, and not necessarily planar. For most realizable choices of angle, the configuration space has both one-dimensional and zero-dimensional components. It becomes zero-dimensional at each end of the realizable range of angles.
+
+- “The configuration space $\mathcal{M}_6(\theta)$ of $\theta$-equiangular unit equilateral hexagons ($\theta \neq \pi/3$) is homeomorphic to a point if $\theta = 0, 2\pi/3$, the union of two circles and four points if $0 < \theta < \pi/3$, and the empty set if $\theta < 0$ or $\theta > 2\pi/3$” (O’Hara, Corollary 3.2).
+- “The configuration space $\mathcal{M}_6(\pi/3)$ of equilateral and $\pi/3$-equiangular hexagons is homeomorphic to the union of a pair of points and the space $X$ illustrated in Fig. 11 [see source] which is a 1-skeleton of a tetrahedron with edges being doubled” (O’Hara, Corollary 3.5).
+
+#### Notes
+
+According to Dan Piker, the configuration space of right-angled equilateral heptagons also has at least one one-dimensional component.
+
 ## Hierarchical constraints
 
 These problems impose various kinds of *soft constraints* on top of the *hard constraints* that an assembly must satisfy to qualify as a solution. Here are some possible kinds of soft constraints.
@@ -28,12 +49,13 @@ These problems impose various kinds of *soft constraints* on top of the *hard co
 - **Optimizing constraints:** Loss function terms whose sum should be minimized.
 - **Optional constraints:** Constraints that may be relaxed if they can't be satisfied.
 
-### Frugal Firepower (optimizing)
+### Frugal firepower (optimizing)
 
 #### Source
 
 - **Author:** David Seppala-Holtzman
-- **Published:** P370 from [the Playground in *Math Horizons*: Volume 25, Issue 4](https://digitaleditions.sheridan.com/article/The+Playground/3039776/483313/article.html)
+- **Published:** P370 from [the Playground in *Math Horizons* 25 (4)](https://doi.org/10.1080/10724117.2018.1434293)
+- **Access:** Openly available from [Sheridan Digital Editions](https://digitaleditions.sheridan.com/article/The+Playground/3039776/483313/article.html)
 
 #### Statement