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Tony PhillipsTony Phillips' Take on Math in the Media
A monthly survey of math news

This month's topics:

Mathematics education in the New York Times

New equilateral polyhedra

A paper in PNAS (February 25, 2014) by Stan Schein and James Gayed (UCLA) explains their discovery of a fourth class of convex equilateral polyhedra with polyhedral symmetry. (Equilateral means that all the edges have the same length; polyhedral symmetry means that each of them has the symmetries of one of the Platonic solids: tetrahedron, octahedron or icosahedron). The first three classes are the Platonic solids themselves, the 13 Archimedean solids and "the two rhombic polyhedra reported by Johannes Kepler in 1611" --the rhombic dodecahedron and triacontahedron. As in Kepler's examples, the faces of the polyhedra in this fourth class (the convex, equilateral Goldberg polyhedra) are polygons that are planar, equilateral but not in general equiangular.

The authors start with Goldberg's 1937 method for generating combinatorial "cages" starting from the three Platonic solids with triangular faces: tetrahedron, octahedron and icosahedron. By overlaying in different ways an equilateral triangle on a regular hexagonal grid they obtain three families of hexagonal patterns in the triangle. One example is the pattern they call (3,0). They print this pattern on each (triangular) face of one of the solids; for example the isocahedron. Where segments of the pattern meet across a polyhedron edge, they are replaced by straight-line segments in 3-space.

triangle with hexagonal pattern icosahedron with (3,0) faces

Construction of a cage. The (3,0) pattern is printed on each face of an icosahedron; the broken segments crossing edges are replaced by straight-line segments in 3-space, shown thickened and in grey here. Images from PNAS 111 no. 8, 2920-2925, used with permission.

In this example pulling the icosahedron faces slightly apart will make the grey lines as long as the others, so the cage becomes equilateral (image below). But if we try to make each of the hexagons and pentagons equiangular as well, solid geometry forces many of those faces to be non-planar, so the cage is not a convex polyhedron. The authors pinpoint and quantify the problem as follows.

dihedral angle

In a geometric convex polyhedron (requires planar faces) suppose an edge, like the red edge in this illustration, joins two trivalent vertices where the face angles are $\alpha, \beta, \gamma$, and $\alpha', \beta', \gamma'$. Note that $\alpha$ and $\alpha'$ are on the faces opposite the edge.

The dihedral angle $A$ along the edge (the angle in space between the two planar faces it joins) can be calculated at either vertex from the equation $$\cos A = \frac{\cos\alpha-\cos\beta\cdot\cos\gamma}{\sin\beta\cdot\sin\gamma} = \frac{\cos\alpha'-\cos\beta'\cdot\cos\gamma'} {\sin\beta'\cdot\sin\gamma'}.$$ (Paul Kunkel has posted a derivation of this equation from the law of cosines).

Schein and Gayed define the dihedral angle discrepancy (DAD) along an edge of the cage to be the difference of the "dihedral" values calculated from the face angles at one end and at the other. If all the polygons in the icosahedral cage are equiangular, then for the edge joining a pentagon to a hexagon all the adjacent face angles are $120^{\circ}$ except for the one in the pentagon, which is $108^{\circ}$. The dihedral calculation along that edge gives $180^{\circ}$ at the 3-hexagon vertex, and $\arccos(-.7453..) = 138.2^{\circ}$ at the other. This shows that the two hexagons bordered by that edge cannot both be planar. Furthermore the DAD gives a measure of the non-planarity, namely $41.8^{\circ}$ in this instance.

The authors finish their construction by proving that if the hexagon angles are allowed to vary (while preserving overall icosahedral symmetry) there are just enough degrees of freedom to make all of the DAD's zero and therefore to produce a unique symmetric, convex, equilateral polygon of the same combinatorial type as the cage. In the final count there is one such polyhedron built on the tetrahedral model, one on the octahedral, and a countable infinity of examples based on the icosahedron.

equilateral cage convex polyhedron

The equilateral cage is not convex: this is apparent along the edges. But there is a (unique) way to adjust the angles in the hexagons so as to make it a convex polyhedron while keeping the full icosahedral symmetry. Larger images: equilateral cage, convex polyhedron. Thanks to Stan Schein for these images.

This work was picked up by Dana Mackenzie in Science News, March 22, 2014, with the title: "Goldberg variations: New shapes for molecular cages."

Multimillion-Dollar Minds

"The Multimillion-Dollar Minds of 5 Mathematical Masters," by Kenneth Chang, ran in the New York Times on June 23, 2014. The piece reports the award of $3 million "Breakthrough" prizes to five mathematicians: Simon Donaldson (my colleague at Stony Brook), Maxim Kontsevich (IHÉS), Jacob Lurie (Harvard), Terence Tao (UCLA) and Richard Taylor (IAS). The prizes are financed by Yuri Milner, "a Russian who dropped out of graduate studies in physics and became a successful investor in Internet companies like Facebook," and Mark Zuckerberg, the founder of Facebook. As Chang tells us, "The Breakthrough Prize in Mathematics is the latest effort in Mr. Milner's crusade to make science lucrative and cool in a society that much more often celebrates athletes, entertainers, politicians and business tycoons." (There are also prizes in Physics and in the Life Sciences). Chang does his best to give some idea of the mathematics being rewarded.

But he tosses in the towel for Lurie: "Dr. Lurie was cited for cutting-edge advances in esoteric fields like 'higher category theory' and 'derived algebraic geometry'."

Tony Phillips
Stony Brook University
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