And the ratios of their numbers, motions, and other properties, everywhere God, as far as necessity allowed or gave consent, has exactly perfected, and harmonised in due proportion. We must imagine all these to be so small that no single particle of any of the four kinds is seen by us on account of their smallness: but when many of them are collected together their aggregates are seen. Let it be agreed, then, both according to strict reason and according to probability, that the pyramid is the solid which is the original element and seed of fire and let us assign the element which was next in the order of generation to air, and the third to water. Of all these elements, that which has the fewest bases must necessarily be the most moveable, for it must be the acutest and most penetrating in every way, and also the lightest as being composed of the smallest number of similar particles : and the second body has similar properties in a second degree, and the third body in the third degree. Also we assign the smallest body to fire, and the greatest to water, and the intermediate in size to air and, again, the acutest body to fire, and the next in acuteness to, air, and the third to water. Wherefore, in assigning this figure to earth, we adhere to probability and to water we assign that one of the remaining forms which is the least moveable and the most moveable of them to fire and to air that which is intermediate. Now, of the triangles which we assumed at first, that which has two equal sides is by nature more firmly based than that which has unequal sides and of the compound figures which are formed out of either, the plane equilateral quadrangle has necessarily, a more stable basis than the equilateral triangle, both in the whole and in the parts. To earth, then, let us assign the cubical form for earth is the most immoveable of the four and the most plastic of all bodies, and that which has the most stable bases must of necessity be of such a nature. 360 BCE), assigning each of them to one of the elements. This profundity captured the imagination of Plato, who gave the solids a fundamental role in his dialogue Timaeus (c. The Platonic solids are somewhat profound, in that there are a small number of them, and no (obvious) reason why they have the number of faces that they do. Obviously, this is ideal for making a fair die - there is no preferred edge, vertex or face on the solid. To put it another way: a Platonic solid can be rotated to make any edge, vertex or face look exactly like any other one. A “regular” polyhedron is one for which not only are all faces equivalent to one another, but so are all edges and all vertices (points). The Platonic solids are the only polyhedra (multi-sided objects) which are convex (have no concavities) and regular. The Platonic solids, taken from an old set of Dungeons & Dragons dice. The Platonic solids include the d4, d6, d8, d12, and d20, as shown below. The most symmetric shapes, as their name implies, formally date back to the Greek philosopher Plato (428-348 BCE), though most of them were recognized, or at least crafted, even earlier. The most obvious way to do that is to make the die have a lot of symmetry in its shape, which brings us to our first category… Before we begin, we should note that a major consideration for any type of die is that it be “fair”: that is, every number on the die should be equally likely to be rolled. We will start with the most familiar types of dice, and work ourselves gradually into strange and unfamiliar territory. I thought it would be fun to answer these questions with a blog post, in which we discuss the geometry of dice! This variety got me wondering: how does one design dice with a weird number of faces? What mathematical strategies does one use to make them? What other types of dice are possible? And, perhaps most important: are these dice “fair”? This is a really amazing variety of dice! This is my “special dice” collection in its entirety, and includes some duplicates (don’t ask how I got 4 identical d60s), but in some cases, such as the d7, d12 and d24, there are varieties in shapes even with the same number of faces! Most human beings never go beyond ordinary 6-sided dice, which we in the gaming world call a “d6.” Classic Dungeons & Dragons players, however, are familiar with the d4, d6, d8, d10, d12, and d20.īut these days, there are even more imaginative varieties! I’ve starting collecting dice of every shape and size, and my current collection is shown below, in order¹: One thing that has changed dramatically since my gaming days is the proliferation of types of dice. I’ve been enjoying a bit of reminiscing about my childhood lately, hunting down old copies of role-playing games I enjoyed in my youth as well as exploring newer games that have come out since then.
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