## Spikeys Everywhere

We call it “Spikey”, and in my life today, it’s everywhere:

It comes from a 3D object—a polyhedron that’s called a rhombic hexecontahedron:

But what is its story, and how did we come to adopt it as our symbol?

## The Origins of Spikey

Back in 1987, when we were developing the first version of Mathematica, one of our innovations was being able to generate resolution-independent 3D graphics from symbolic descriptions. In our early demos, this let us create wonderfully crisp images of Platonic solids. But as we approached the release of Mathematica 1.0, we wanted a more impressive example. So we decided to take the last of the Platonic solids—the icosahedron—and then make something more complex by a certain amount of stellation (or, more correctly, cumulation). (Yes, that’s what the original notebook interface looked like, 30 years ago…)

At first this was just a nice demo that happened to run fast enough on the computers we were using back then. But quite soon the 3D object it generated began to emerge as the de facto logo for Mathematica. And by the time Mathematica 1.0 was released in 1988, the stellated icosahedron was everywhere:

In time, tributes to our particular stellation started appearing—in various materials and sizes:

But just a year after we released Mathematica 1.0, we were getting ready to release Mathematica 1.2, and to communicate its greater sophistication, we wanted a more sophisticated logo. One of our developers, Igor Rivin, had done his PhD on polyhedra in hyperbolic space—and through his efforts a hyperbolic icosahedron adorned our Version 1.2 materials:

My staff gave me an up-to-date-Spikey T-shirt for my 30th birthday in 1989, with a quote that I guess even after all these years I’d still say:

After Mathematica 1.2, our marketing materials had a whole collection of hyperbolic Platonic solids, but by the time Version 2.0 arrived in 1991 we’d decided our favorite was the hyperbolic dodecahedron:

Still, we continued to explore other “Spikeyforms”. Inspired by the “wood model” style of Leonardo da Vinci’s stellated icosahedron drawing (with amazingly good perspective) for Luca Pacioli’s book De divina proportione, we commissioned a Version 2.0 poster (by Scott Kim) showing five intersecting tetrahedra arranged so that their outermost vertices form a dodecahedron:

Looking through my 1991 archives today, I find some “explanatory” code (by Ilan Vardi)—and it’s nice to see that it all just runs in our latest Wolfram Language (though now it can be written a bit more elegantly):

Over the years, it became a strange ritual that when we were getting ready to launch a new integer version of Mathematica, we’d have very earnest meetings to “pick our new Spikey”. Sometimes there would be hundreds to choose from, generated (most often by Michael Trott) using all kinds of different algorithms:

But though the color palettes evolved, and the Spikeys often reflected (though perhaps in some subtle way) new features in the system, we’ve now had a 30-year tradition of variations on the hyperbolic dodecahedron:

In more recent times, it’s become a bit more streamlined to explore the parameter space—though by now we’ve accumulated hundreds of parameters:

A hyperbolic dodecahedron has 20 points—ideal for celebrating the 20th anniversary of Mathematica in 2008. But when we wanted something similar for the 25th anniversary in 2013 we ran into the problem that there’s no regular polyhedron with 25 vertices. But (essentially using `SpherePoints[25]`) we managed to create an approximate one—and made a 3D printout of it for everyone in our company, sized according to how long they’d been with us:

## Enter Wolfram|Alpha

In 2009, we were getting ready to launch Wolfram|Alpha—and it needed a logo. There were all sorts of concepts:

We really wanted to emphasize that Wolfram|Alpha works by doing computation (rather than just, say, searching). And for a while we were keen on indicating this with some kind of gear-like motif. But we also wanted the logo to be reminiscent of our longtime Mathematica logo. So this led to one of those classic “the-CEO-must-be-crazy” projects: make a gear mechanism out of Spikey-like forms.

Longtime Mathematica and Wolfram Language user (and Hungarian mechanical engineer) Sándor Kabai helped out, suggesting a “Spikey Gear”:

And then, in a throwback to the Version 2 intersecting tetrahedra, he came up with this:

In 2009, 3D printing was becoming very popular, and we thought it would be nice for Wolfram|Alpha to have a logo that was readily 3D printable. Hyperbolic polyhedra were out: their spikes would break off, and could be dangerous. (And something like the Mathematica Version 4 Spikey, with “safety spikes”, lacked elegance.)

For a while we fixated on the gears idea. But eventually we decided it’d be worth taking another look at ordinary polyhedra. But if we were going to adopt a polyhedron, which one should it be?

There are of course an infinite number of possible polyhedra. But to make a nice logo, we wanted a symmetrical and somehow “regular” one. The five Platonic solids—all of whose faces are identical regular polygons—are in effect the “most regular” of all polyhedra:

Then there are the 13 Archimedean solids, all of whose vertices are identical, and whose faces are regular polygons but of more than one kind:

One can come up with all sorts of categories of “regular” polyhedra. One example is the “uniform polyhedra”, as depicted in a poster for The Mathematica Journal in 1993:

Over the years that Eric Weisstein was assembling what in 1999 became MathWorld, he made an effort to include articles on as many notable polyhedra as possible. And in 2006, as part of putting every kind of systematic data into Mathematica and the Wolfram Language, we started including polyhedron data from MathWorld. The result was that when Version 6.0 was released in 2007, it included the function `PolyhedronData` that contained extensive data on 187 notable polyhedra:

It had always been possible to generate regular polyhedra in Mathematica and the Wolfram Language, but now it became easy. With the release of Version 6.0 we also started the Wolfram Demonstrations Project, which quickly began accumulating all sorts of polyhedron-related Demonstrations.

One created by my then-10-year-old daughter Catherine (who happens to have continued in geometry-related directions) was “Polyhedral Koalas”—featuring a pull-down for all polyhedra in `PolyhedronData[]`:

So this was the background when in early 2009 we wanted to “pick a polyhedron” for Wolfram|Alpha. It all came to a head on the evening of Friday, February 6, when I decided to just take a look at things myself.

I still have the notebook I used, and it shows that at first I tried out the rather dubious idea of putting spheres at the vertices of polyhedra:

But (as the Notebook History system recorded) just under two minutes later I’d generated pure polyhedron images—all in the orange we thought we were going to use for the logo:

The polyhedra were arranged in alphabetical order by name, and on line 28, there it was—the rhombic hexecontahedron:

A couple of minutes later, I had homed in on the rhombic hexecontahedron, and at exactly 12:24:24am on February 7, 2009, I rotated it into essentially the symmetrical orientation we now use:

I wondered what it would look like in gray scale or in silhouette, and four minutes later I used `ColorSeparate` to find out:

I immediately started writing an email—which I fired off at 12:32am:

“I [...] rather like the RhombicHexecontahedron ….

It’s an interesting shape … very symmetrical … I think it might have

about the right complexity … and its silhouette is quite reasonable.”

I’d obviously just copied “RhombicHexecontahedron” from the label in the notebook (and I doubt I could have spelled “hexecontahedron” correctly yet). And indeed from my archives I know that this was the very first time I’d ever written the name of what was destined to become my all-time-favorite polyhedron.

It was dead easy in the Wolfram Language to get a picture of a rhombic hexecontahedron to play with:

✕
PolyhedronData["RhombicHexecontahedron"] |

And by Monday it was clear that the rhombic hexecontahedron was a winner—and our art department set about rendering it as the Wolfram|Alpha logo. We tried some different orientations, but soon settled on the symmetrical “head-on” one that I’d picked. (We also had to figure out the best “focal length”, giving the best foreshortening.)

Like our Version 1.0 stellated icosahedron, the rhombic hexecontahedron has 60 faces. But somehow, with its flower-like five-fold “petal” arrangements, it felt much more elegant. It took a fair amount of effort to find the best facet shading in a 2D rendering to reflect the 3D form. But soon we had the first official version of our logo:

It quickly started to show up everywhere, and in a nod to our earlier ideas, it often appeared on a “geared background”:

A few years later, we tweaked the facet shading slightly, giving what is still today the logo of Wolfram|Alpha:

## The Rhombic Hexecontahedron

What is a rhombic hexecontahedron? It’s called a “hexecontahedron” because it has 60 faces, and ἑξηκοντα (hexeconta) is the Greek word for 60. (Yes, the correct spelling is with an “e”, not an “a”.) It’s called “rhombic” because each of its faces is a rhombus. Actually, its faces are golden rhombuses, so named because their diagonals are in the golden ratio ≃1.618:

The rhombic hexecontahedron is a curious interpolation between an icosahedron and a dodecahedron (with an icosidodecahedron in the middle). The 12 innermost points of a rhombic hexecontahedron form a regular icosahedron, while the 20 outermost points form a regular dodecahedron. The 30 “middle points” form an icosidodecahedron, which has 32 faces (20 “icosahedron-like” triangular faces, and 12 “dodecahedron-like” pentagonal faces):

Altogether, the rhombic hexecontahedron has 62 vertices and 120 edges (as well as 120−62+2=60 faces). There are 3 kinds of vertices (“inner”, “middle” and “outer”), corresponding to the 12+30+20 vertices of the icosahedron, icosidodecahedron and dodecahedron. These types of vertices have respectively 3, 4 and 5 edges meeting at them. Each golden rhombus face of the rhombic hexecontahedron has one “inner” vertex where 5 edges meet, one “outer” vertex where 3 edges meet and two “middle” vertices where 4 edges meet. The inner and outer vertices are the acute vertices of the golden rhombuses; the middle ones are the obtuse vertices.

The acute vertices of the golden rhombuses have angle 2 tan^{−1}(*ϕ*^{−1}) ≈ 63.43°, and the obtuse ones 2 tan^{−1}(*ϕ*) ≈ 116.57°. The angles allow the rhombic hexecontahedron to be assembled from Zometool using only red struts (the same as for a dodecahedron):

Across the 120 edges of the rhombic hexecontahedron, the 60 “inward-facing hinges” have dihedral angle 4𝜋/5=144°, and the 60 “outward-facing” ones have dihedral angle 2𝜋/5=72°. The solid angles subtended by the inner and outer vertices are 𝜋/5 and 3𝜋/5.

To actually draw a rhombic hexecontahedron, one needs to know 3D coordinates for its vertices. A convenient way to get these is to use the fact that the rhombic hexecontahedron is invariant under the icosahedral group, so that one can start with a single golden rhombus and just apply the 60 matrices that form a 3D representation of the icosahedral group. This gives for example final vertex coordinates {±*ϕ*,±1,0}, {±1,±*ϕ*,±(1+*ϕ*)}, {±2*ϕ*,0,0}, {±*ϕ*,±(1+2*ϕ*),0}, {±(1+*ϕ*),±(1+*ϕ*),±(1+*ϕ*)}, and cyclic permutations of these, with each possible sign being taken.

In addition to having faces that are golden rhombuses, the rhombic hexecontahedron can be constructed out of 20 golden rhombohedra (whose 6 faces are all golden rhombuses):

There are other ways to build rhombic hexecontahedra out of other polyhedra. Five intersecting cubes can do it, as can 182 dodecahedra with touching faces:

Rhombic hexecontahedra don’t tessellate space. But they do interlock in a satisfying way (and, yes, I’ve seen tens of paper ones stacked up this way):

There are also all sorts of ring and other configurations that can be made with them:

Closely related to the rhombic hexecontahedron (“RH”) is the rhombic triacontahedron (“RT”). Both the RH and the RT have faces that are golden rhombuses. But the RH has 60, while the RT has 30. Here’s what a single RT looks like:

RTs fit beautifully into the “pockets” in RHs, leading to forms like this:

The aforementioned Sándor Kabai got enthusiastic about the RH and RT around 2002. And after the Wolfram Demonstrations Project was started, he and Slovenian mathematician Izidor Hafner ended up contributing over a hundred Demonstrations about RH, RT and their many properties:

## Paper Spikey Kits

As soon as we’d settled on a rhombic hexecontahedron Spikey, we started making 3D printouts of it. (It’s now very straightforward to do this with `Printout3D[PolyhedronData[...]]`, and there are also precomputed models available at outside services.)

At our Wolfram|Alpha launch event in May 2009, we had lots of 3D Spikeys to throw around:

But as we prepared for the first post-Wolfram|Alpha holiday season, we wanted to give everyone a way to make their own 3D Spikey. At first we explored using sets of 20 plastic-covered golden rhombohedral magnets. But they were expensive, and had a habit of not sticking together well enough at “Spikey scale”.

So that led us to the idea of making a Spikey out of paper, or thin cardboard. Our first thought was then to create a net that could be folded up to make a Spikey:

My daughter Catherine was our test folder (and still has the object that was created), but it was clear that there were a lot of awkward hard-to-get-there-from-here situations during the folding process. There are a huge number of possible nets (there are already 43,380 even for the dodecahedron and icosahedron)—and we thought that perhaps one could be found that would work better:

But after failing to find any such net, we then had a new (if obvious) idea: since the final structure would be held together by tabs anyway, why not just make it out of multiple pieces? We quickly realized that the pieces could be 12 identical copies of this:

And with this we were able to create our “Paper Sculpture Kits”:

Making the instructions easy to understand was an interesting challenge, but after a few iterations they’re now well debugged, and easy for anyone to follow:

And with paper Spikeys in circulation, our users started sending us all sorts of pictures of Spikeys “on location”:

## The Path to the Rhombic Hexecontahedron

It’s not clear who first identified the Platonic solids. Perhaps it was the Pythagoreans (particularly living near so many polyhedrally shaped pyrite crystals). Perhaps it was someone long before them. Or perhaps it was a contemporary of Plato’s named Theaetetus. But in any case, by the time of Plato (≈400 BC), it was known that there are five Platonic solids. And when Euclid wrote his Elements (around 300 BC) perhaps the pinnacle of it was the proof that these five are all there can be. (This proof is notably the one that takes the most steps—32—from the original axioms of the Elements.)

Platonic solids were used for dice and ornaments. But they were also given a central role in thinking about nature, with Plato for example suggesting that perhaps everything could in some sense be made of them: earth of cubes, air of octahedra, water of icosahedra, fire of tetrahedra, and the heavens (“ether”) of dodecahedra.

But what about other polyhedra? In the 4th century AD, Pappus wrote that a couple of centuries earlier, Archimedes had discovered 13 other “regular polyhedra”—presumably what are now called the Archimedean solids—though the details were lost. And for a thousand years little more seems to have been done with polyhedra. But in the 1400s, with the Renaissance starting up, polyhedra were suddenly in vogue again. People like Leonardo da Vinci and Albrecht Dürer routinely used them in art and design, rediscovering some of the Archimedean solids—as well as finding some entirely new polyhedra, like the icosidodecahedron.

But the biggest step forward for polyhedra came with Johannes Kepler at the beginning of the 1600s. It all started with an elegant, if utterly wrong, theory. Theologically convinced that the universe must be constructed with mathematical perfection, Kepler suggested that the six planets known at the time might move on nested spheres geometrically arranged so as to just fit the suitably ordered five Platonic solids between them:

In his 1619 book Harmonices mundi (“Harmony of the World”) Kepler argued that many features of music, planets and souls operate according to similar geometric ratios and principles. And to provide raw material for his arguments, Kepler studied polygons and polyhedra, being particularly interested in finding objects that somehow formed complete sets, like the Platonic solids.

He studied possible “sociable polygons”, that together could tile the plane—finding, for example, his “monster tiling” (with pentagons, pentagrams and decagons). He studied “star polyhedra” and found various stellations of the Platonic solids (and in effect the Kepler–Poinsot polyhedra). In 1611 he had published a small book about the hexagonal structure of snowflakes, written as a New Year’s gift for a sometime patron of his. And in this book he discussed 3D packings of spheres (and spherical atoms), suggesting that what’s now called the Kepler packing (and routinely seen in the packing of fruit in grocery stores) is the densest possible packing (a fact that wasn’t formally proved until into the 2000s—as it happens, with the help of Mathematica).

There are polyhedra lurking in Kepler’s various packings. Start from any sphere, then look at its neighbors, and join their centers to make the vertices of a polyhedron. For Kepler’s densest packing, there are 12 spheres touching any given sphere, and the polyhedron one gets is the cuboctahedron, with 12 vertices and 14 faces. But Kepler also discussed another packing, 8% less dense, in which 8 spheres touch a given sphere, and 6 are close to doing so. Joining the centers of these spheres gives a polyhedron called the rhombic dodecahedron, with 14 vertices and 12 faces:

Having discovered this, Kepler started looking for other “rhombic polyhedra”. The rhombic dodecahedron he found has rhombuses composed of pairs of equilateral triangles. But by 1619 Kepler had also looked at golden rhombuses—and had found the rhombic triacontahedron, and drew a nice picture of it in his book, right next to the rhombic dodecahedron:

Kepler actually had an immediate application for these rhombic polyhedra: he wanted to use them, along with the cube, to make a nested-spheres model that would fit the orbital periods of the four moons of Jupiter that Galileo had discovered in 1610.

Why didn’t Kepler discover the rhombic hexecontahedron? I think he was quite close. He looked at non-convex “star” polyhedra. He looked at rhombic polyhedra. But I guess for his astronomical theories he was satisfied with the rhombic triacontahedron, and looked no further.

In the end, of course, it was Kepler’s laws—which have nothing to do with polyhedra—that were Kepler’s main surviving contribution to astronomy. But Kepler’s work on polyhedra—albeit done in the service of a misguided physical theory—stands as a timeless contribution to mathematics.

Over the next three centuries, more polyhedra, with various forms of regularity, were gradually found—and by the early 1900s there were many known to mathematicians:

But, so far as I can tell, the rhombic hexecontahedron was not among them. And instead its discovery had to await the work of a certain Helmut Unkelbach. Born in 1910, he got a PhD in math at the University of Munich in 1937 (after initially studying physics). He wrote several papers about conformal mapping, and—perhaps through studying mappings of polyhedral domains—was led in 1940 to publish a paper (in German) about “The Edge-Symmetric Polyhedra”.

His goal, he explains, is to exhaustively study all possible polyhedra that satisfy a specific, though new, definition of regularity: that their edges are all the same length, and these edges all lie in some symmetry plane of the polyhedron. The main result of his paper is a table containing 20 distinct polyhedra with that property:

Most of these polyhedra Unkelbach knew to already be known. But Unkelbach singles out three types that he thinks are new: two hexakisoctahedra (or disdyakis dodecahedra), two hexakisicosahedra (or dysdyakis triacontahedra), and what he calls the Rhombenhexekontaeder, or in English, the rhombic hexecontahedron. He clearly considers the rhombic hexecontahedron his prize specimen, including a photograph of a model he made of it:

How did he actually “derive” the rhombic hexecontahedron? Basically, he started from a dodecahedron, and identified its two types of symmetry planes:

Then he subdivided each face of the dodecahedron:

Then he essentially considered pushing the centers of each face in or out to a specified multiple *α* of their usual distance from the center of the dodecahedron:

For *α* < 1, the resulting faces don’t intersect. But for most values of *α*, they don’t have equal-length sides. That only happens for the specific case —and in that case the resulting polyhedron is exactly the rhombic hexecontahedron.

Unkelbach actually viewed his 1940 paper as a kind of warmup for a study of more general “k-symmetric polyhedra” with looser symmetry requirements. But it was already remarkable enough that a mathematics journal was being published at all in Germany after the beginning of World War II, and soon after the paper, Unkelbach was pulled into the war effort, spending the next few years designing acoustic-homing torpedoes for the German navy.

Unkelbach never published on polyhedra again, and died in 1968. After the war he returned to conformal mapping, but also started publishing on the idea that mathematical voting theory was the key to setting up a well-functioning democracy, and that mathematicians had a responsibility to make sure it was used.

But even though the rhombic hexecontahedron appeared in Unkelbach’s 1940 paper, it might well have languished there forever, were it not for the fact that in 1946 a certain H. S. M. (“Donald”) Coxeter wrote a short review of the paper for the (fairly new) American Mathematical Reviews. His review catalogs the polyhedra mentioned in the paper, much as a naturalist might catalog new species seen on an expedition. The high point is what he describes as “a remarkable rhombic hexecontahedron”, for which he reports that “its faces have the same shape as those of the triacontahedron, of which it is actually a stellation”.

Polyhedra were not exactly a hot topic in the mathematics of the mid-1900s, but Coxeter was their leading proponent—and was connected in one way or another to pretty much everyone who was working on them. In 1948 he published his book Regular Polytopes. It describes in a systematic way a variety of families of regular polyhedra, in particular showing the great stellated triacontahedron (or great rhombic triacontahedron)—which effectively contains a rhombic hexecontahedron:

But Coxeter didn’t explicitly mention the rhombic hexecontahedron in his book, and while it picked up a few mentions from polyhedron aficionados, the rhombic hexecontahedron remained a basically obscure (and sometimes misspelled) polyhedron.

## Quasicrystals

Crystals had always provided important examples of polyhedra. But by the 1800s, with atomic theory increasingly established, there began to be serious investigation of crystallography, and of how atoms are arranged in crystals. Polyhedra made a frequent appearance, in particular in representing the geometries of repeating blocks of atoms (“unit cells”) in crystals.

By 1850 it was known that there were basically only 14 possible such geometries; among them is one based on the rhombic dodecahedron. A notable feature of these geometries is that they all have specific two-, three-, four- or six-fold symmetries—essentially a consequence of the fact that only certain polyhedra can tessellate space, much as in 2D the only regular polygons that can tile the plane are squares, triangles and hexagons.

But what about for non-crystalline materials, like liquids or glasses? People had wondered since before the 1930s whether at least approximate five-fold symmetries could exist there. You can’t tessellate space with regular icosahedra (which have five-fold symmetry), but maybe you could at least have icosahedral regions with little gaps in between.

None of this was settled when in the early 1980s electron diffraction crystallography on a rapidly cooled aluminum-manganese material effectively showed five-fold symmetry. There were already theories about how this could be achieved, and within a few years there were also electron microscope pictures of grains that were shaped like rhombic triacontahedra:

And as people imagined how these triacontahedra could pack together, the rhombic hexecontahedron soon made its appearance—as a “hole” in a cluster of 12 rhombic triacontahedra:

At first it was referred to as a “20-branched star”. But soon the connection with the polyhedron literature was made, and it was identified as a rhombic hexecontahedron.

Meanwhile, the whole idea of making things out of rhombic elements was gaining attention. Michael Longuet-Higgins, longtime oceanographer and expert on how wind makes water waves, jumped on the bandwagon, in 1987 filing a patent for a toy based on magnetic rhombohedral blocks, that could make a “Kepler Star” (rhombic hexecontahedron) or a “Kepler Ball” (rhombic triacontahedron):

And—although I only just found this out—the rhombohedral blocks that we considered in 2009 for widespread “Spikey making” were actually produced by Dextro Mathematical Toys (aka Rhombo.com), operating out of Longuet-Higgins’s house in San Diego.

The whole question of what can successfully tessellate space—or even tile the plane—is a complicated one. In fact, the general problem of whether a particular set of shapes can be arranged to tile the plane has been known since the early 1960s to be formally undecidable. (One might verify that 1000 of these shapes can fit together, but it can take arbitrarily more computational effort to figure out the answer for more and more of the shapes.)

People like Kepler presumably assumed if a set of shapes was going to tile the plane, they must be able to do so in a purely repetitive pattern. But following the realization that the general tiling problem is undecidable, Roger Penrose in 1974 came up with two shapes that could successfully tile the plane, but not in a repetitive way. By 1976 Penrose (as well as Robert Ammann) had come up with a slightly simpler version:

And, yes, the shapes here are rhombuses, though not golden rhombuses. But with angles 36°,144° and 72°,108°, they arrange with 5- and 10-fold symmetry.

By construction, these rhombuses (or, more strictly, shapes made from them) can’t form a repetitive pattern. But it turns out they can form a pattern that can be built up in a systematic, nested way:

And, yes, the middle of step 3 in this sequence looks rather like our flattened Spikey. But it’s not exactly right; the aspect ratios of the outer rhombuses are off.

But actually, there is still a close connection. Instead of operating in the plane, imagine starting from half a rhombic triacontahedron, made from golden rhombuses in 3D:

Looking at it from above, it looks exactly like the beginning of the nested construction of the Penrose tiling. If one keeps going, one gets the Penrose tiling:

Looked at “from the side” in 3D, one can tell it’s still just identical golden rhombuses:

Putting four of these “Wieringa roofs” together one can form exactly the rhombic hexecontahedron:

But what’s the relation between these nested constructions and the actual way physical quasicrystals form? It’s not yet clear. But it’s still neat to see even hints of rhombic hexecontahedra showing up in nature.

And historically it was through their discussion in quasicrystals that Sándor Kabai came to start studying rhombic hexecontahedra with Mathematica, which in turn led Eric Weisstein to find out about them, which in turn led them to be in Mathematica and the Wolfram Language, which in turn led me to pick one for our logo. And in recognition of this, we print the nestedly constructed Penrose tiling on the inside of our paper Spikey:

## Flattening Spikey

Our Wolfram|Alpha Spikey burst onto the scene in 2009 with the release of Wolfram|Alpha. But we still had our long-running and progressively evolving Mathematica Spikey too. So when we built a new European headquarters in 2011 we had not just one, but two Spikeys vying to be on it.

Our longtime art director Jeremy Davis came up with a solution: take one Spikey, but “idealize” it, using just its “skeleton”. It wasn’t hard to decide to start from the rhombic hexecontahedron. But then we flattened it (with the best ratios, of course)—and finally ended up with the first implementation of our now-familiar logo:

## The Brazilian Surprise

When I started writing this piece, I thought the story would basically end here. After all, I’ve now described how we picked the rhombic hexecontahedron, and how mathematicians came up with it in the first place. But before finishing the piece, I thought, “I’d better look through all the correspondence I’ve received about Spikey over the years, just to make sure I’m not missing anything.”

And that’s when I noticed an email from June 2009, from an artist in Brazil named Yolanda Cipriano. She said she’d seen an article about Wolfram|Alpha in a Brazilian news magazine—and had noticed the Spikey—and wanted to point me to her website. It was now more than nine years later, but I followed the link anyway, and was amazed to find this:

I read more of her email: “Here in Brazil this object is called ‘Giramundo’ or ‘Flor Mandacarú’ (Mandacaru Flower) and it is an artistic ornament made with [tissue paper]”.

What?! There was a Spikey tradition in Brazil, and all these years we’d never heard about it? I soon found other pictures on the web. Only a few of the Spikeys were made with paper; most were fabric—but there were lots of them:

I emailed a Brazilian friend who’d worked on the original development of Wolfram|Alpha. He quickly responded “These are indeed familiar objects… and to my shame I was never inquisitive enough to connect the dots”—then sent me pictures from a local arts and crafts catalog:

But now the hunt was on: what were these things, and where had they come from? Someone at our company volunteered that actually her great-grandmother in Chile had made such things out of crochet—and always with a tail. We started contacting people who had put up pictures of “folk Spikeys” on the web. Quite often all they knew was that they got theirs from a thrift shop. But sometimes people would say that they knew how to make them. And the story always seemed to be the same: they’d learned how to do it from their grandmothers.

The typical way to build a folk Spikey—at least in modern times—seems to be to start off by cutting out 60 cardboard rhombuses. The next step is to wrap each rhombus in fabric—and finally to stitch them all together:

OK, but there’s an immediate math issue here. Are these people really correctly measuring out 63° golden rhombuses? The answer is typically no. Instead, they’re making 60° rhombuses out of pairs of equilateral triangles—just like the standard diamond shapes used in quilts. So how then does the Spikey fit together? Well, 60° is not far from 63°, and if you’re sewing the faces together, there’s enough wiggle room that it’s easy to make the polyhedron close even without the angles being precisely right. (There are also “quasi-Spikeys” that—as in Unkelbach’s construction—don’t have rhombuses for faces, but instead have pointier “outside triangles”.)

Folk Spikeys on the web are labeled in all sorts of ways. The most common is as “Giramundos”. But quite often they are called “Estrelas da Felicidade” (“stars of happiness”). Confusingly, some of them are also labeled “Moravian stars”—but actually, Moravian stars are different and much pointier polyhedra (most often heavily augmented rhombicuboctahedra) that happen to have recently become popular, particularly for light fixtures.

Despite quite a bit of investigation, I still don’t know what the full history of the “folk Spikey” is. But here’s what I’ve found out so far. First, at least what survives of the folk Spikey tradition is centered around Brazil (even though we have a few stories of other appearances). Second, the tradition seems to be fairly old, definitely dating from well before 1900 and quite possibly several centuries earlier. So far as I can tell—as is common with folk art—it’s a purely oral tradition, and so far I haven’t found any real historical documentation about it.

My best information has come from a certain Paula Guerra, who sold folk Spikeys at a tourist-oriented cafe she operated a decade ago in the historic town of São Luíz do Paraitinga. She said people would come into her cafe from all over Brazil, see the folk Spikeys and say, “I haven’t seen one of those in 50 years…”

Paula herself learned about folk Spikeys (she calls them “stars”) from an older woman living on a multigenerational local family farm, who’d been making them since she was a little girl, and had been taught how to do it by her mother. Her procedure—which seems to have been typical—was to get cardboard from anywhere (originally, things like hat boxes), then to cover it with fabric scraps, usually from clothes, then to sew the whole perhaps-6″-across object together.

How old is the folk Spikey? Well, we only have oral tradition to go by. But we’ve tracked down several people who saw folk Spikeys being made by relatives who were born around 1900. Paula said that a decade ago she’d met an 80-year-old woman who told her that when she was growing up on a 200-year-old coffee farm there was a shelf of folk Spikeys from four generations of women.

At least part of the folk Spikey story seems to center around a mother-daughter tradition. Mothers, it is said, often made folk Spikeys as wedding presents when their daughters went off to get married. Typically the Spikeys were made from scraps of clothes and other things that would remind the daughters of their childhood—a bit like how quilts are sometimes made for modern kids going to college.

But for folk Spikeys there was apparently another twist: it was common that before a Spikey was sewn up, a mother would put money inside it, for her daughter’s use in an emergency. The daughter would then keep her Spikey with her sewing supplies, where her husband would be unlikely to pick it up. (Some Spikeys seem to have been used as pincushions—perhaps providing an additional disincentive for them to be picked up.)

What kinds of families had the folk Spikey tradition? Starting around 1750 there were many coffee and sugar plantations in rural Brazil, far from towns. And until perhaps 1900 it was common for farmers from these plantations to get brides—often as young as 13—from distant towns. And perhaps these brides—who were typically from well-off families of Portuguese descent, and were often comparatively well educated—came with folk Spikeys.

In time the tradition seems to have spread to poorer families, and to have been preserved mainly there. But around the 1950s—presumably with the advent of roads and urbanization and the move away from living on remote farms—the tradition seems to have all but died out. (In rural schools in southern Brazil there were however apparently girls in the 1950s being taught in art classes how to make folk Spikeys with openings in them—to serve as piggy banks.)

Folk Spikeys seem to have shown up with different stories in different places around Brazil. In the southern border region (near Argentina and Uruguay) there’s apparently a tradition that the “Star of St. Miguel” (aka folk Spikey) was made in villages by healer women (aka “witches”), who were supposed to think about the health of the person being healed while they were sewing their Spikeys.

In other parts of Brazil, folk Spikeys sometimes seem to be referred to by the names of flowers and fruits that look vaguely similar. In the northeast, “Flor Mandacarú” (after flowers on a cactus). In tropical wetland areas, “Carambola” (after star fruit). And in central forest areas “Pindaíva” (after a spiky red fruit).

But the most common current name for a folk Spikey seems to be “Giramundo”—an apparently not-very-recent Portuguese constructed word meaning essentially “whirling world”. The folk Spikey, it seems, was used like a charm, and was supposed to bring good luck as it twirled in the wind. The addition of tails seems to be recent, but apparently it was common to hang up folk Spikeys in houses, perhaps particularly on festive occasions.

It’s often not clear what’s original, and what’s a more recent tradition that happens to have “entrained” folk Spikeys. In the Three Kings’ Day parade (as in the three kings from the Bible) in São Luiz do Paraitinga, folk Spikeys are apparently used to signify the Star of Bethlehem—but this seems to just be a recent thing, definitely not indicative of some ancient religious connection.

We’ve found a couple of examples of folk Spikeys showing up in art exhibitions. One was in a 1963 exhibition about folk art from northeastern Brazil organized by architect Lina Bo Bardi. The other, which happens to be the largest 3D Spikey I’ve ever seen, was in a 1997 exhibition of work by architect and set designer Flávio Império:

So… where did the folk Spikey come from? I still don’t know. It may have originated in Brazil; it may have come from Portugal or elsewhere in Europe. The central use of fabrics and sewing needed to make a “60° Spikey” work might argue against an Amerindian or African origin.

One modern Spikey artisan did say that her great grandmother—who made folk Spikeys and was born in the late 1800s—came from the Romagna region of Italy. (One also said she learned about folk Spikeys from her French-Canadian grandmother.) And I suppose it’s conceivable that at one time there were folk Spikeys all over Europe, but they died out enough generations ago that no oral tradition about them survives. Still, while a decent number of polyhedra appear, for example, in European paintings from earlier centuries, I don’t know of a single Spikey among them. (I also don’t know of any Spikeys in historical Islamic art.)

But ultimately I’m pretty sure that somewhere there’s a single origin for the folk Spikey. It’s not something that I suspect was invented more than once.

I have to say that I’ve gone on “art origin hunts” before. One of the more successful was looking for the first nested (Sierpiński) pattern—which eventually led me to a crypt in a church in Italy, where I could see the pattern being progressively discovered, in signed stone mosaics from just after the year 1200.

So far the Spikey has proved more elusive—and it certainly doesn’t help that the primary medium in which it appears to have been explored involved fabric, which doesn’t keep the way stone does.

## Spikeys Come to Life

Whatever its ultimate origins, Spikey serves us very well as a strong and dignified icon. But sometimes it’s fun to have Spikey “come to life”—and over the years we’ve made various “personified Spikeys” for various purposes:

When you use Wolfram|Alpha, it’ll usually show its normal, geometrical Spikey. But just sometimes your query will make the Spikey “come to life”—as it does for pi queries on Pi Day:

## Spikeys Forever

Polyhedra are timeless. You see a polyhedron in a picture from 500 years ago and it’ll look just as clean and modern as a polyhedron from my computer today.

I’ve spent a fair fraction of my life finding abstract, computational things (think cellular automaton patterns). And they too have a timelessness to them. But—try as I might—I have not found much of a thread of history for them. As abstract objects they could have been created at any time. But in fact they are modern, created because of the conceptual framework we now have, and with the tools we have today—and never seen before.

Polyhedra have both timelessness and a rich history that goes back thousands of years. In their appearance, polyhedra remind us of gems. And finding a certain kind of regular polyhedron is a bit like finding a gem out in the geometrical universe of all possible shapes.

The rhombic hexecontahedron is a wonderful such gem, and as I have explored its properties, I have come to have even more appreciation for it. But it is also a gem with a human story—and it is so interesting to see how something as abstract as a polyhedron can connect people across the world with such diverse backgrounds and objectives.

Who first came up with the rhombic hexecontahedron? We don’t know, and perhaps we never will. But now that it is here, it’s forever. My favorite polyhedron.

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