Industrious Dice |

I’m a professional mathematician. That nastys somebody pays me to do math. I’m also a recreational mathematician. That nastys you might have to pay me to get me to stop.

Wearing my recreational mathematician hat – quite literpartner, as you’ll see – I gave a talk earlier this year on some newfangled dice that do the same sorts of jobs as ancigo in-createed dice but with scanter pips (“pips” being the technical name for all those little bdeficiency dots).

Fewer pips? Why would anyone on earth attfinish about that? Dice manufacturers certainly don’t; drilling the pips and coloring them in isn’t a convey inant expense. But I’m not an applied mathematician. I’m a sanitize/recreational mathematician, and I’m helped to attfinish about anyskinnyg I pick to. When I pick well, I discover someskinnyg new and engaging.

FEWER PIPS

The standard die (called a d6 in the gaming community) has one job: to create a random number between 1 and 6. The d6 accomplishes this job in a straightforward create by having 1, 2, 3, 4, 5, and 6 pips on its six faces, using a total of 1 + 2 + 3 + 4 + 5 + 6 = 21 pips; becaengage of the cube’s symmetries, each of the six faces has an equivalent chance of being the face that’s facing up when the die stops rolling.

But here’s a top watch of a die that gets the same job done with far scanter pips:

This die is not a 6-sided firm but a 12-sided firm called a dodecahedron. To engage this newfangled die, throw it, paengage till it stops rolling, and count not equitable the pips on the top face but also the pips on the five surrounding faces; that is, count the pips on the six faces that are clear from straightforwardly above the die. For instance, in the picture above we see six pips, so the throw would have appreciate 6. The pips are scheduled in such a create that the appreciate of a throw is equpartner probable to be 1, 2, 3, 4, 5, or 6.

When you roll such a die repeatedly, each face of the dodecahedron is an upper face one-half of the time. In contrast, when you roll a cubical die repeatedly, each face of the cube is the top face only one-sixth of the time. So the dodecahedral die is (from a pip-engage perspective) three times as effective as the cubical die (one-half is three times hugeger than one-sixth), and it correplyingly does its job with only one-third as many pips: 7 rather than 21.

To portray such a die, I had to mend a confengage: Put non-adverse integers into the twelve regions in the diagram below so that the twelve sums you get by compriseing each number to its five neighbors are 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 in some order. (See Endremark #1 for one solution.) All but one of the regions are pentagons; the last one (the outer region) watchs enjoy a square with a pentagonal hole in it. To get you begined, I’ve put the number 0 into the outer region.

FEWER DIMENSIONS

The join between my die-portray problem and my fill-in-the-numbers confengage goes by way of a create of polyhedral cartography innovateed by the 19th century German mathematician Victor Schlegel. He was a conmomentary of the more expansively-understandn mathematician and authorr Charles Howard Hinton who coined the now-well-understandn word “tesseract” to portray a four-unreasonableensional analogue of the cube. Hinton and Schlegel labored in an era when the geometry of higher-unreasonableensional space had seized the imaginations of mathematicians and non-mathematicians aenjoy. Schlegel’s fantastic contribution was discovering a way to suppress n-unreasonableensional hyperfirms (or maybe equitable gentle them a wee bit) by projecting them down into (n−1)-unreasonableensional space. Such diagrams can help us steer higher-unreasonableensional objects. We’ll intensify on the case n = 3 and see how Schlegel would have had us map the faces of the 3-unreasonableensional dodecahedron in the 2-unreasonableensional set upe.

To picture Schlegel’s trick, envision that it’s night-time in a up-to-dateistic children’s take partground that features a dodecahedral climbing arrange (someskinnyg Hinton’s son Sebastian would have probably enjoyd since Sebastian Hinton is now reassembleed as the inventor of the jungle gym). It’s not a firm dodecahedron but an airy assemblage of struts (thirty, if you attfinish to count them). Imagine that you climb on top of the dodecahedron and shine a flashweightless down on the arrange besystematich you, from a point equitable above the middle of the arrange’s pentagonal top face; the shadows cast on the ground below create the Schlegel diagram.

(Here I’ve engaged denseness of the line segments to correply to how shut the admireive struts are from the watcher’s perspective, or roverdelightedly, how expansive the shadows are.)

FINDING A STORY TO TELL

This dodecahedral die is an outincreaseth of the Muñiz-Propp die I alludeed in my earlier essay “Let x equivalent x“. I choosed a year ago that I wanted to give a talk on these two dice at an event held this past February called the Gathering for Gardner; it’s a biennial conclave of mathematicians, magicians, and other folks who adore the labor of Martin Gardner, and since both mathematicians and magicians enjoy dice, the topic seemed enjoy a authentic fit.

But then I thought, wouldn’t it be even better to do someskinnyg aenjoy involving dice with fifteen sides, since the 2024 encountering was the fifteenth of its comardent?

It isn’t the first time I’ve given a talk at a Gathering for Gardner that saluted the ordinal number of that particular Gathering; for instance, I setd a talk for the fourteenth Gathering based on Stan Wagon’s terrific article “Fourteen Proofs of a Result About Tiling a Rectangle”. That particular Gathering had to be rescheduled becaengage of the COVID-19 pandemic, and I finished up not conshort-terming a talk at the rescheduled (and virtual) fourteenth Gathering for Gardner, but I still wrote a Mathematical Enchantments essay based on the talk I’d nastyt to give: “What Proof Is Best?”

Unblessedly, there’s a problem with fifteen-sided dice. If you go to the web in search of a “d15” die, you’ll discover they all come with the cautioning “This die is not isohedral” (or words to that effect). Isohedral polyhedra are polyhedra with the pleasant property that for any two faces F and G of the polyhedron, there’s a symmetry of the polyhedron that carries F to G. Cubes are isohedral, and so are dodecahedra. Isohedral convex polyhedra create excellent dice becaengage they’re equitable as probable to land on any face as on any other. But there are no n-sided isohedral convex polyhedra when n is an odd number (enjoy the number 15).

On the other hand, there are 30-sided isohedral convex polyhedra, and indeed one of them – the triacontahedron – is the basis of the standard d30 die. Since 30 is twice 15, it’d be basic to engage such a die to create a random number between 1 and 15: have two faces shothriveg each of the numbers from 1 to 15. But of course I don’t want to engage numbers on the faces; the whole point is to engage pips, and to engage them as “effectively” as possible. Of course we could engage 1+1+2+2+…+15+15 = 240 pips and equitable count the pips on the top face, but who wants a die that has 15 pips on some of its faces? Either the pips would be minuscule or the die would be huge. Instead, when we roll such a die, we should count the pips on the top face and the four neightedious faces. This incrrelieves the efficiency by a factor of five, so the total number of pips is equitable 240/5 or 48 — assuming, that is, that such an schedulement exists.

As in the case of my dodecahedral die for simulating a d6, the problem of arranging pips on a triacontahedral die to simutardy a d15 can be recast as the confengage of filling numbers in a Schlegel digram so that all the possible sums we want (in this case, all the numbers between 1 and 15) are equpartner reconshort-termed. But now there are 30 variables instead of equitable 12, and I establish that my brute-force approach to solving the confengage, which had thriveed for the dodecahedron, was too sluggish to produce even a individual solution. So I accomplished out for help.

As it happens, I’m not the only person I understand who walks the line between sanitize math and recreational math – and who, in the process of walking the line, tries to blur it a little. For cut offal decades Rich Schroeppel has run an email forum called math-fun for folks enjoy me who don’t draw a acute distinction between the two comardents of math; such mathematical luminaries as Bill Thurston and John Conway were active in the group in its timely days. Two current members are George Hart and Robin Houston, and they were able to discover cut offal solutions to my confengage.

Among the solutions they establish, one stands out as especipartner prenting. If you want to see what their die watchs enjoy, print out the follothriveg image on some stiff paper, cut out the netlabor of 30 rhombengages, fancigo in aextfinished the edges uniteing adjacent rhombengages, and tape the whole skinnyg up to create a triacontahedron.

Especipartner pleasing to me are the follothriveg three properties that this die satisfies.

(1) No face has more than 3 pips. I don’t want too many pips to have to allot a face – that’d force the pips to be petite or the die to be huge.

(2) If you count the pips clear from overhead and the pips clear from undersystematich (imagining that the die rests on a glass table), you always get a total of 16 pips. This property is reminiscent of the way the two opposite faces of a cubical die show numbers that comprise up to 7.

(3) There are equivalent numbers of faces with 1 pips, 2 pips, and 3 pips. This property lfinishs more visual variety to the die.

But what I repartner enjoy is that Houston and Hart showed that these three properties, apshown together, individual out the die shown above. As a mathematician, I’m enticeed to systems of constraints that are daintyly poised between being too lax and too recut offeive. A problem that has only one solution is an especipartner prenting comardent of problem to come atraverse or to invent.

TELLING THE STORY I FOUND

Once I knew there was a new and nifty way to create a random number between 1 and 15, I had the seed for my Gathering for Gardner 15 (“G4G15“) talk. In set upning the talk, I choosed to dramatize my quirky portray criterion by creating and inhabiting a pompous character who would attfinish about the problem much more than I did, and who would watch the efficiency of the die not equitable as an esthetic publish but a moral one.

You’ll see that this character wears a top hat. Although I borrowed the hat from one of the magicians uniteing the Gathering, it was intfinished to be more enjoy a capitaenumerate’s hat than a magician’s hat (skinnyk of the hat in the board game Monopoly). More particularpartner, he’s a merciless industriaenumerate who’s offfinished that the pips on a standard die don’t labor challenging enough.

I don’t skinnyk the triacontahedral d15 has much of a future in the gaming world, despite the existence of a petite but excited community of gamers who assemble novelty dice. Nor does the dodecahedral d6. This griefful conclusion is a consequence of the geometry of gaming in the physical world. Think about a game enjoy Monopoly that engages dice: you have two or more take parters sitting around a board and all take parters demand to be able to see what appreciate each take parter has rolled. The more effective (or, as my character would say, “firmlaboring”) a die is, the skinnyer the range of watching-angles is for the take parters who want to verify what appreciate was rolled, since each take parter must see all the relevant faces at once. Two people sitting on opposite sides of a board wouldn’t be able to see the top five faces of the dodecahedral d6 at the same time; they’d have to apshow turns watching it from above. The same goes for people using a triacontahedral d15. Imagine two take parters trying to get a watch of the die at the same time. Hilarity or romance might ensue, but the possibility of skull trauma cannot be diswatchd. So: manufacture and engage firmlaboring dice at your own hazard!

Still, the process of coming up with these dice recommended me some of the same satisfactions as doing a piece of “solemn” research, so I don’t watch it as time misengaged. And if my talk on the topic plrelieved a scant hundred people, all the better.

Here are a couple of concluding confengages for you:

First, presume we chamfer the twelve edges of a blank die and put pips on the chamfered edges; then when we roll the die, four edges are always on top (the four edges that bound the top face). Can we schedule pips on the edges so that when we roll the die and count the pips on the top four edges, we get a random number between 1 and 6? Find such an schedulement or show that none exists. See Endremark #2.

Second, presume we truncate the eight corners instead and put pips on the truncated corners; then when we roll the die, four corners are always on top (the four corners of the top face). Can we schedule pips at the corners so that when we roll the die and count the pips on the top four corners, we get a random number between 1 and 6? Find such an schedulement or show that none exists. See Endremark #3.

Thanks to George Hart and Robin Houston without whose help I couldn’t have given the talk and wouldn’t have written this essay. Great thanks to the schedulers of the 15th Gathering for Gardner. Thanks also to Tom Roby for recommending that I wear a top hat and to The Time-Traveling Wizard (aka Roger Manderscheid) for providing one.

ENDNOTES

#1. Here’s one solution:

On the left I show a solution to the confengage; on the right I show the twelve sums that verify that the solution satisfies the demandment of produceing each sum between 1 and 6 exactly twice.

#2. No such die exists. Increatepartner, we can argue that since each pip on such a die “counts” (that is, gives to the appreciate of the throw) one-third of the time (unenjoy the pips of a standard die which count only one-sixth of the time), the recommendd die must be exactly twice as effective as a standard die, nastying the number of pips must be not 21 but 21/2. But 21/2 is not an integer. More rigorously, we could portrayate twelve variables a,b,…,l to reconshort-term the number of pips on the twelve chamfered edges. For each of the six possible ways the die could land, the number of pips shown by the surrounding edges is a sum of four of the variables. If we comprise up all six of these sums, we get 2a+2b+…+2l, where each variable occurs with the coeffective 2 becaengage each edge beextfinisheds to exactly two faces. But the six sums are presumed to comprise up to 1+2+3+4+5+6, or 21; and 2a+2b+…+2l is even whereas 21 is odd, so no such portrayatement of pips exists.

#3. Many such dice exist. My preferite is one in which no corner-face has more than 2 pips (a reasonable skinnyg to demand since the corner-faces are not very huge). Here is what the portrayatement of pips watchs enjoy on a Schlegel diagram of the cube:

If one creates the corner-faces as huge as possible, the result is a firm called a cuboctahedron with 8 triangular faces and 6 square faces. One could create a cuboctahedral d6 that has 7 pips allotd among the triangular faces, but a take parter might have to roll such a die repeatedly until it lands on a square face.

Hmm, is a thrown cuboctahedron enjoylier to finish up resting on a triangular face or on a square face? A square face, I skinnyk; but how much enjoylier? If any of you have a sturdy cuboctahedron lying around and do some experiments, let me understand what you discover!