(2012-11-26) How many different ways to mark a die,if pips on opposite faces must add up to 7?
Answer : 16. (It's 240 if opposite faces needn't add up to 7.)
My local bargain store sells two types of dices from China,for 99¢ a pack.
Five transparent red cubes with white pips, 19 mm on a side.
Twelve 15 mm cubes (white, green & red) with black or white pips.
Besides the obvious difference in sizes and colors, I noticed that thedices in the two packs were not alike because of the arrangementof the spots.
(2012-11-29) Let's drop the rule that requires opposite faces to add up to 7.
If the three faces 2-3-6 share one vertex (which they always do in a traditional die) then they can be arranged in two configurations (left-handed and right-handed). For either of those, there are 6 ways to arrange the "hidden" faces 1,4,5 (in only one of those is the sum of opposing faces always equal to 7). That accounts for a total of 12 possible configurations (so far).
If the 2-3-6 faces do not meet at a vertex, then two of them must be opposingeach other (with the third sharing an edge with the other two). There are just three possible such configuration (think of it as choosing which of the 2-3-6 is the middle one). With respect to any such base, the other three faces can be arranged in 6 different configurations. That's a total of 18 possible configurations. Adding that to the 12 configurations described in the previous paragraph,we obtain a grand total of 30 possible configurations.
Configurations in which opposing faces never add up to 7 :
For either of the two ways the 2-3-6 faces can meet at a vertex,we have to put either 4 or 5 opposite to 6, then we have no further leeway. That accounts for 4 configurations (so far).
For any of the three ways two of the 2-3-6 faces are opposing each other,we have two choices for the face opposite to the "middle" face and two furtherchoices for setting the remaining two faces (that are opposite to each other). That's 12 configurations to add to the 4 enumeratedin the previous paragraph, for a grand total of 16.
All told, in the 30 ways to number a dice, there are two ways (right-handedand left-handed) in which opposing sides always add up to seven and 16 ways in which they never do.
If its pyramidal components aren't too flat,such a 3n-hedron is only stable on an horizontalplane if its rests on one of its n lateral faces. If the diehas an order-n axis of symmetry, all those faces are equiprobable because they'reequivalent (assuming uniform mass density).
(2012-11-28) Rock-paper-scissors (Rochambo) with dice.
Here's a nontransitive set of three 3-sided dice (or 6-sided dice with the same spots on opposite sides) due to Dr. Nathaniel Hellerstein, CCSF :
Red = {3,5,7}. Yellow = {2,4,9}. Blue = {1,6,8}. With probability 5/9 in every case, red beats yellow, yellow beats blue and blue beats red:
Red beats Yellows
3
5
7
2
4
9
Yellow beats Blue
1
6
8
3
5
7
Blue beats Red
2
4
9
1
6
8
Transitivity, or lack thereof...
Transitivity is a fundamental property of ordering relations which formally statesthat if A ranks B and B ranks C, then A must rank C.
We're so used to ranking things transitively that not being able to do so is disturbing. None of the above three dice is preferable to both of the others. There are many other entertaining examples with dice.
A far more serious nontransitive case is democratic majority voting. Indeed, it's possible that voters who would prefer A to B and B to C wouldactually prefer C to A. That's Condorcet's paradox. An unfortunate fact of life.
Scams based on nontransitivity will fool people who don't know what to look for. One nice example is the so-called Penny Ante game which Walter Francis Penney (1913-2000) introduced in October 1969 (in 10 lines). Martin Gardner (1914-2010) discussed it in the MathematicalGames column of the October 1974 issue ofScientific American :
Each player predicts that a sequence of three heads (1) or tails (0) will occur in a sequenceof flips of a fair coin before what the other has predicted. That game would be fair if both predictions were pickedat random (as if inscribed on two balls drawn from an urn with 8 balls). However, the opportunity to pick aprediction knowing the choice of the first player always gives the second playeran opportunity for a 2:1 advantage or better. As is often the case, an informed choice is better than a random one.
Probability of a second-player win in Penney's game with optimal strategy
Prediction of Player A
000
001
010
011
100
101
110
111
Best Prediction for Player B
100
100
001
001
110
110
011
011
Winning Probability for B
7/8
3/4
2/3
2/3
2/3
2/3
3/4
7/8
To justify the above optimal strategy, we have to compute the probability of a win for all possible responses of B to all choices of A. This is best done using an algorithm due to John Conway (1937-2020) presented in the paper of Humble and Nishiyamaquoted in the footnotes below. Conway's algorithm applies to the generalized Penneygame, where the two predictions need not be of length 3 (they don't even have to be of the same length).
With predictions of the same length k≥4 János Csirik (1946-) found in 1992 that Player A's best choice to limit the advantage of B is a sequence whose bitsare all indentical except 3 (1 at the beginning and 2 at the end). This leaves a option with winning odds (1+2k-1):(1+2k-2).
"Optimal strategy for the first player in the Penney ante game" by J.A. Csirik, Combinatorics, Probability and Computing, Volume 1, Issue 4 (1992), pp 311-321.
Steve Humble,MBE and Yutaka Nishiyama (1948-) prefer to play Penney Ante with a deck of cards instead (26 black cards and 26 red ones). Be that as it may, the exact probabilities involved in the "Humble-Nishiyama randomnsess game"differ slightly from those of the original Penney game and they're tougher to work out, unless a computer is used. In certain cases, there's even a nonzero probability of a tie (undecided game) which can't happen in Penney Ante with an unlimited number of coin flips. For example, when A bets "001" and B wisely replies with "100" the game is tied if "00" never occurs, which happens if and onlyif red and black always keep alternating, as they do with a probability of:
2 / C(52,26) = 4.03292... 10-15 which is minute but nonzero.
(2012-11-28) Any total, from 2 to 12, has the same probability as with standard dice.
One die has pips 1,2,2,3,3,4 and the other is marked 1,3,4,5,6,8.
These dice were invented by Colonel George L. Sicherman (then of Buffalo, New York) whose discovery was reported by Martin Gardner, in one of his legendary Scientific American columns (1978).
The best way to investigate this matter involvesgenerating polynomials. Besides proving the basic claim, this approach can establish the uniqueness of Sicherman's dice among 6-sided dice with nonzero markings:
Proof :
To a face with n spots, we assign the monomial xn. To the whole die correspond the sum of the polynomials associated with its faces. For example, the polynomial associated to a standard die is:
S = x + x2 + x3 + x4 + x5 + x6
The number of ways we can obtain a total of n pips when we roll severaldice is the coefficient of xn in the product oftheir polynomials (: to obtain thatterm, you must sum up all the ways there are to pick one term from each factorso that the exponents of x add up to n).
Therefore, with two standard dice (a red one and a green one, say) the number of ways to roll a total of n pips is the coefficientof xn in the square of the abovepolynomial. Namely:
Now, the interesting remark is that S can be factored:
S = x ( 1 + x ) ( 1 - x + x2) ( 1 + x + x2)
We may regroup the factors of the square S2 in the following way:
S2 = [x ( 1 + x ) ( 1 + x + x2)] [x ( 1 + x ) ( 1 - x + x2)2 ( 1 + x + x2)]
The two square brackets expand respectively as follows:
x + 2x2 + 2x3 + x4 = x + x2 + x2 + x3 + x3 + x4 and x + x3 + x4 + x5 + x6 + x8
Those correspond to 6-sided dice marked 1,2,2,3,3,4 and 1,3,4,5,6,8.
What's somewhatmiraculous is that we end up witha pair of 6-sided dice. To match what's done with traditional dice, those dice should be builtwith opposite faces adding up to 5 for the lower die and 9 for the upper die.
Other groupings of the above factors of S2 yield proper dice only when every resulting polynomial has nonnegative coefficents. We obtain:
1-2-4-5 tetrahedron with 1-2-3-3-4-5-5-6-7 enneahedron.
1-4 coin and 1-2-2-3-3-3-4-4-4-5-5-5-6-6-6-7-7-8 octadecahedron.
(2013-02-05) When does the sum of two dice give equiprobable totals?
The standard set of seven polyhedral dice made popular by Dungeons & Dragonsconsists of the fiveplatonicsolids and a pair of 10-sided pentagonaldeltohedra. One is marked from 0 to 9 and the other from 00 to 90. Those two are known as percentile dice. When rolled together, the percentile dice give any total from 0 to 99 with equal probability (1/100). In traditional role playing games (RPG) a total of zero (0+00) is interpreted as 100.
More generally, we may consider a set of p n-sided fair dicewhere the j+1st face of the i+1st die is marked j.ni ). When those dice are rolled, they giveany total from 0 to np-1 with equal probability. Let's generalize:
For a prescribed integer M, what are the setsfair dice marked integers that will give any total between 0 and M-1 with probability 1/M ?
Well, the polynomial approach introduced in theprevious sectionreduces this question to the factorization of the polynomial:
(1-xM)/ (1-x) = 1 + x + x2 + x3 + ... + xM-1
The factors of those polynomials are called cyclotomic ("cycle-splitting") and they've been studied and cataloged by generations of mathematicians.
Dismissing as trivial the type of splitting described in the above introduction, the firstnon-trivial factorization is for M = 6:
A factorization gives a legitimate set of dice only if all the factors are polynomials whosecoefficients are nonnegative integers. In this case, only three possibilities exist:
A single six-sided die marked (0,1,2,3,4,5).
A 3-sided (curved) die marked (0,1,2) and a coin marked (0,3).
A coin marked (0,1) and a 3-sided (curved) die marked (0,2.4).
More generally, we can devise such a set of marked dice for any ordered factorization of the integer M. If M is prime, there's only one solution (a single die with M sides).
(2013-04-14) Dice people actually use, for divination or recreation.
Rôle playing games (RPG) call fora variety of dice besides the traditional 6-sided cubic dice (D6). The most popular sets have 7 dice:
Icosahedra were already used in Antiquity, for divination purposes. The large (52 mm) glass dieshown at left is one of the most famous extant examples (c. AD 100). It was auctioned off at Christie's for $17925 on December 11, 2003.
Prior to that auction, it had drawn little attention andwas expected to fetch between $4000 and $6000. It would be worth a lot more now.
For 12-sided dice, the regular dodecahedron has eclipsed the rhombic dodecahedron, which was apparently mass-produced only once, around 1978, for the AskAstro-Dice fortune-telling game.
For the shot at left, I got an old set from eBay ($15 on 2013-03-29). New onesuse regular dodecahedra, unfortunately...
Note that Pluto (top symbol on the center die) wasstill a planet back then.
In recent years, two distinct isohedra with 24 faceshave been mass-produced as dice by Louis Zocchi (hear Lou's pitch). One is the isohedral tetrakis hexahedron or tetrakis cube pictured at left. The other is a large die in the shape of a deltoidal icositetrahedron(strombic icositetrahedron or trapezoidal icositetrahedron ). It's marketed by GameScience (Zocchi's company) under the name of D-Total, featuring fancy markingsthat are meant to facilitate the use of the die as a substitute for dice with 2, 3, 4, 5, 6, 7, 8, 10, 12, 20, 24, 30, 40, 50, 60, 70 or 80 sides. This is jointly credited to Dr. Alexander F. Simkin, Frank Dutrain (of LD Diffusion) and Louis Zocchi (2009).
(2013-04-26) Some of the most common sizes for cubic dice are:
5 mm micro,
8 mm tiny,
12 mm mini ( less than 1/2'' )
15 mm regular ( 5/8'' standard RPG size, in the US )
16 mm medium ( largest backgammon size )
19 mm large ( 3/4'' casino size )
25 mm or 28 mm jumbo ( 1'' )
35 mm or 38 mm giant ( 1½'' )
55 mm monster ( 2'' or more )
Polyhedral dice are loosely matched with 6-sided dice of similar bulk:
Caliper Sizes (between any face and its opposite element)
Mini 12 mm
Regular 15 mm
Large 19 mm
Jumbo 28 mm
Giant 35 mm
D4
17.4 mm
D6
12 mm
15.0 mm
19 mm
25 mm
D8
14.8 mm
24 mm
D10
15.9 mm
29 mm
D12
18.5 mm
26 mm
D20
19.3 mm
D24
19.5 mm
24 mm
D30
32 mm
Oversized dice could damage dice trays. They're best tossed on carpets.
(2013-04-14) All the faces of an isohedron are equivalent.
An isohedron is a polyhedron whoses faces are all equivalent. That's to say that every face can be transformed into any other through some spatial isometry (rotation or reflection) that maps the polyhedron onto itself.
Thedual of an isohedron is an isogonal polyhedron,and vice-versa (duality being understoodwith respect to the sphere inscribed in the isohedron or circumscribed to its dual). Sometimes, the dual is more readily understood than the primal.
The dual of a convex solid (polyhedralor not) is convex. So, every convex isogonal polyhedron is associated to an isohedron and vice-versa. In particular, every Catalan solid (i.e., the dual of one of the 13Archimedean polyhedra) is an isohedron. So are the duals of isogonal prisms and antiprisms (respectively called amphihedra and deltohedra).
However, there's no requirement that the dual of an isohedron be equilateral. So, there are isohedra that are not duals of uniform polyhedra (as uniform means both isogonal and equilateral).
For example, the familiarregular tetrahedronis not the only isohedral tetrahedron... Any tetrahedron whose opposing edges have the same length (as illustrated at right) is an isohedron. Such a tetrahedron is called a disphenoid. The dual of a disphenoid is another disphenoid;disphenoids are both isogonal and isohedral (they're thus noble).
Therefore, any disphenoid would make a perfectly fair 4-sided die. A disphenoid is chiraliff the three quantities a,b and c are distinct.
(2013-02-22) A steel ball moves in an isogonal cavity, dual of the isohedral outer pattern.
Currently, "6-sided" spherical dice areavailable (the inner cavity is octahedral ).
Any isohedral diecould be made this way, by carving out a cavityin the shape of its dual (the dual of an isohedron is an isogonal polyhedron ). However, the fairness of such a die would only be guaranteed if the cavitywas isohedral as well. In other words, it must be noble (that word simply means both isogonal and isohedral).
Among convex polyhedra, the only noble ones are the Platonic solids and the disphenoids. The latter type would yield a great way to make 4-sided dice without any sharp corners (in fact, without any corners at all). If a scalene disphenoid is used, the artefact would be a sphere wherethe outer markings would seem asymmetrically distributed. Yet, it would be a perfectly fair die and, therefore,a great conversation piece...
(2013-02-14) There's only one such solid (besides the 6-antiprisms and 12-prisms).
If d is the degree of every vertex in an isogonal polyhedron, it has at most d different edge lengths and d non-congruent faces. When both maxima are achieved, theisogonal polyhedron is said to be scalene.
Consider now the three distinct types of isogonal tetradecahedra obtained by cutting off the eight corners of a cube with increasing severity:
In the first two cases, the truncation must be the same at all corners of the cube, or else we wouldn't obtain a polyhedron with isogonal symmetry.
However, if the truncation planes intersect inside the cube (along a new edge) then the isogonal symmetry persistswhen different truncations are used for nonadjacentvertices of the cube. The resulting scalene isogonal tetradecahedron is azonohedron featuring 3 edge lengths,6 rectangular faces and two distinct types of hexagonal faces. At right is my own old school cardboard modelwith voided rectangular faces, where edges are in a 1:2:3 proportion. (This photo may fool the eye if you're not aware that surfaces with pencil patternsare inner ones.)
By definition, an isohedron is said to be scalene if itsdual is (the dual of an isohedronis an isogonal polyhedron and vice-versa). For example, the dual of the polyhedron just described isa scalene isohedron (a fair die) with 24 faces congruent to the sametriangle (scalenetetrakis hexahedron).
(Gerard Villarreal, TX. 2013-02-05) What polyhedron is this 14-sided Korean die ?
The juryeonggu is precisely an isogonaltruncated octahedron.
Because its tetragonal faces are square,it's a special case of theabove solid. The shape is fully specified bythe length (x) of the edges between hexagons,assuming the square faces have unit sides.
For x = 1, we would obtain the uniform truncated octahedron (pictured above at left) which features regular hexagonal faces. (Don't call it a cuboctahedron, which isanother uniform tetradecahedron, as is thetruncated cube.) ThisArchimedean solid can tile space without voids (a non-uniform juryeonggu can't). It's the basis for thenear-optimal foam describedbyLord Kelvin in 1887 (Kelvin's cell has the same vertices as the tetradecahedron but all its edges are curvedand so are its hexagonal faces).
For any x, a juryeonggu has the following dimensions (derivedbelow). The locution "radiusto" denotes the distance from the polyhedron's center. The radius to any vertex is the radius R of the circumscribed sphere.
Dimensions of an Isogonal Truncated Octahedron (Korean Juryeonggu)
Definition & Symbolic Name
Value
Side of a square face (basic unit)
a4
1
Length of an edge between hexagonal faces
a6
x
Diagonal of a square face
2
Diagonal of an hexagonal face
1 + x
Width of an hexagon (between parallel sides)
3/2 ( 1 + x )
Radius of the circumscribed sphere
R
( 1 + x + ½ x2)½
Radius of a square face
r4
2/2
Radius of an hexagonal face
r6
3/3 ( 1 + x + x2)½
Radius to [center of] a square face
h4
2/2 ( 1 + x )
Radius to [center of] an hexagonal face
h6
6/3 ( 1 + ½ x )
Caliper ratio ( h4 / h6)
3 (1+x) / (2+x)
Radius to [centers of] sides of squares
R4
½ ( 3 + 4x + 2x2)½
Radius to [centers of] other edges
R6
1 + x/2
Solid angle of a square face
4
4 Arcsin [ 1 / (3+4x+2x2)]
Solid angle of an hexagonal face
6
/2- 3 Arcsin (3+4x+2x2)-1
Surface area of a square face
A4
1
Surface area of an hexagonal face
A6
3/4 (1 + 4x + x2)
Total surface area ( 6A4+8A6)
A
6 + 23 (1+4x+x2)
Volume of a square pyramid ( A4 h4 / 3)
V4
2/6 ( 1 + x )
Volume of an hexagonal pyramid ( A6 h6 / 3)
V6
2/24(1+4x+x2)(2+x)
Total Volume
V
6 V4 + 8 V6
Derivations for the above (outline) :
R can be obtained as the radius of the equator circumscribed to an isogonal octagon whose sides are eitherthe diagonal of a square face (length 2) or an edgeof length x. In that equatorial plane, we alsofind h4 and R6 :
R2 = [ 1 + (1+x)2]/ 2 = 1 + x + ½ x2 h4 = ( 1 + x ) / 2 R6 = 1 + x/2
An hexagonal face is obtained bysubtracting three equilateral triangles of side x from an equilateral triangle of side 1+2x,so its surface area is:
The diagonal of an isogonal hexagon with sides 1 and x is (1+x). The width between parallelsides is equal to that diagonal multiplied by the sine of 60°. The least obvious quantity is r6 which can be obtained fromplanar cartesian coordinates (that's what we use when all else seems to fail). We then obtain h6 from the Pythagorean theorem:
R2 = r62 + h62
R4 is the height of an isosceles triangle of base 1 with two sides equal to R.
R42 = R2 - ¼ = ( 3 + 4x + 2x2) / 4
We may check that R4 = R6 in the uniform case (x = 1). The solid angle subtended by a square face is obtained immediately from R4 using the formula we've establishedelsewhereon this site. For hexagonal faces, we just usethe fact that the solid angles subtended by all faces add up to 4.
Looking for the Perfect Juryeonggu :
The official blog of the city of Gyeongju states that, in a traditional juryeonggu, all faces have the same surface area. This entails a quadratic equation in x, whose positive root is:
x = ( 3+4/3 ) 2 = 0.304213765421624907891...
Gerard Villarreal (private communication) advocates a juryeonggu with h4 = h6 so that all faces are tangent to the same inner sphere (which may then be called theinscribed sphere,by analogy with theisohedral concept). This entails a quadratic equation whose positive solution is:
x = (3 - 1) / 2 = 0.36602540378443864676372317...
In either case, it was guessed thatendowing an isogonal truncated octahedron with a particularproperty that isohedra possess would endow them with the same fairness asisohedral dice. It ain't quite so...
Oversimplifying the conclusions of thediscussion below,the latter guess turns out to correspond to the situationwhere the die bounces a very large number of times. At the other extreme is a die that doesn't bounce at all (think of a randomly oriented die immersed in glycerol an dropped just above asticky surface). Such a die would simply land on any face with a probability proportionalto thesolid angle subtended by that facefrom the center of the polyhedron. All faces subtend the same solid angle (2/7) when:
x = (((1 / 2sin/14)-½) - 1 = 0.32173356003298450750124...
Since x can't be both 0.3660 and 0.3217 (obviously) no isogonal truncated octahedron can be unconditonally fair (like an isohedron would be). However, for any given set of physical conditions and casting style,there's one isogonal truncated octahedron that looks fair...
Empirically Fair Dice :
As dice are actually rolled in specific conditions that are somewhere betweenthe two (contrived) extremes described above, a juryeonggu that looks fair in practicewould have to correspond to a parameter x determined empirically underthose given conditions.
One way to do so is to build two isogonal truncated octahedra with different parameters x1 and x2 (not too far from a guess of 0.35).
Cast both 700 times and count how many times they land on an hexagonal face (N1 and N2 respectively). By linear interpolation, we'd approach a perfect score of 400 for a die having thefollowing value of x:
x =
(N2- 400) x1 + (400 - N1) x2
N2 - N1
The two most interesting dice to build are the ones mentioned above:
If you build a third die using the above interpolation, you may roll your three dicemany times and plot with good precision the curve giving the probability of an hexagon as a function of x (with just 3 known points, you may as well assume the curve is a parabola ora circle). Use that information to build a fourth die, if you must. That last die may not be quite fair (with your own particulars) but it's unlikely that anybody will ever detect that!
The precision of the above method is limited by the standard deviation on N, which is about 13 for nearly-perfect dice with 700 trials (it's proportional to the square root of the number of trials).
I've carved a 23.9 mm orthohedral (x = 0.366) juryeonggu to a precision of 0.05 mm and obtained the results tabulated below. They show that an hexagonal face is about twice as likely as a square one, although an hexagon subtends only 9.4% more solid angle than a square.
Orthohedral Juryeonggu on Felt-Covered Wooden Dice Tray
Outcome
100-roll test runs
Total
Per face
Hexagon
72
72
76
70
77
70
80
64
581
= 12.6
72.6
Square
28
28
24
30
23
30
20
36
219
36.5
2 = A.B / (A+B)
(2013-03-27) Discussing non-isohedral commercially available dice.
For lack of symmetry among their faces, the fairness of non-isohedraldice may depend critically on the way they are cast and/or on the resilienceof the landing surface. Such dice may roll true on plexiglass but not on felt (say) or the imparted spin (long roll or not) may bias them.
Nevertheless, those things can substitute for fair dice with odd numbers of sides (all isohedra have an even number of faces). A good example is the 7-sided die pictured above (16.3 mm thick, as devised by Lou Zocchi) whichI found to roll true under typical conditions (two dice tossed together 392 times from a ribbed cup onto a circular felt-covered wooden tray).
For non-isohedral dice to give at least the illusion of rolling true,they must be designed bytrial and error and tested extensively. Sometimes, the necessary rigor isn't exercised in commercial endeavors...
The worst mass-produced offender is probably the 5-sided die (D5) pictured at left, a flawed design for which Lou Zocchi was grantedUS Patent 6926275 in 2005.
In his patent application, Zocchi stated that the devicehad been tested by rolling it 10163 times on plexiglassand was found to be practically fair under such conditions. I find this pretty hard to believe after the following field-test:
Using my trusted ribbed cup and felt-covered dice tray, I cast 5 such dice together 100 times (for a total of 500 individual outcomes) and saw the dice land 282 times onone of the two triangular faces and only 218 times on one of the three rectangular sides. So, the 28.2 % share of each triangular face was nearly twice the 14.5 % share of each rectangular face. Bad.
If needed in actual gaming, a fair D5 die can be nicely simulated by rolling a good D6 cubeuntil the outcome isn't 6 ("casino" dice are machined fromextruded cellulose acetateto a precision better than0.0005"). The average number N of actual rolls required for a valid outcome is only 1.2, since :
N = 1 + N/6 yields N = 6/5
Other good alternatives exist which forego extra rolls entirely,including the use of a 5-sidedspindleor a 10-sided isohedron with duplicate labels.
The celebrated Zocchihedron pictured at left is an interestingrandomizing device but it's not a solid die at all. It can't be rigorously tested as proper dice can because it has memory... It contains a spherical cavity partially filled with sand which helps it cometo a full stop but also makes it impossible to grasp its true dynamical statefrom its outer appearance.
(2013-01-27) Isohedra are intrinsically fair dice. Are there any others ?
Fair dice do not create randomness or uniform probability distributions;they merely conserve it. When thrown from a truly random orientation, a fair die is equally likely to land onany face.
A sufficient condition for an homogeneous convex polyhedron tobe a fair is to be isohedral, for the above is then satisfied by reason of symmetry.
An isohedron is a face-transitive polyhedron. This is to say that every face is the image of any other in at least one isometrictransformation of the entire polyhedron (i.e., a rotation or a mirror reflectionmapping the polyhedron onto itself).
Isohedral symmetry is precisely what guarantees that all facesare strictly equivalent. In any type of damped motion (including, but not limited to, inelastic shocks with fixed objects of any shape) if all initial orientations of an homogeneousrigidisohedral die are equiprobable,then it will certainly come to rest on any on its faces equiprobably.
Revisit now the argument we used for isogonal truncated octahedra to prove that one value of the single parameter describing those shapes must correspond to a fair die which isn't isohedral (no truncated octahedron can possibly be isohedral, since squares and hexagons can't be congruent).
That theoretical argument (and/or the practical recipe we gave to determine something close to thecorrect shape using linear interpolation) was based on the implicit assumptionthat dice are always cast the same way (on an horizontal table covered with a particular shock-absorbing material, say).
(2013-02-07) What two extreme ways of casting dice imply for fair dice.
The thesis [master's thesis |pdf] filed in 1997 by Ed Pegg, Jr. for hisMaster's degree atUCCSwas entitled A Complete List of Fair Dice. In it, the famous recreational mathematician actually classified isohedra. He was fully aware of the remote possibilitythat some fair dice might exist besides isohedra (which are fair by symmetry) but he clearly estimated (rightly so) that the less technicalterm was more suitable for a title. Elsewhere, he answered thequestion "Can a non-isohedral fair die exist?" by using the example of a square pyramid with isosceles lateral faces, thrown as a die under some set of standardconditions. The four triangular faces have the probability by symmetry. The probability of the square face is above that if the pyramid is tall and below that if the pyramid is short. Therefore, anintermediate height mustexist for which all faces have the same probability. He adds:
However, once the conditions changed, the die would no longer be fair. (I have a strong argument for this, but no proof.)
What follows can serve as the proof that Ed Pegg, Jr. is calling for. The main difficulty was that the lack of fairness of something like theaforementioned square pyramid can only be established if we analyzetheoretically at least two sets of physical conditions. The trick is to consider two limiting cases rather than anyrealistic motion. A fair die could theoreticallybe thrown in any possible and couldn't show a differencein probabilities between those two limits, which turn out tobe simple enough to analyze.
(2013-02-07) Slow dead-cat bounce on a sticky surface.
In 1981,David Singmaster (b. 1939) discussed the proposalthat an homogeneous die would land on a face with a probabilityproportional to thesolid angle subtendedby that face (as seen from the center of gravity).
At first sight, the idea looks silly... For one thing, this would assign nonzero probabilities to unstable faces (wheneverthe orthogonal projection of the center of gravity on the plane of the face isn't inside the face,the die cannot rest on that face at all). Also, it would seem to overestimate the probabilities of lateral faces in flat prisms (our physical intuition is that a coin can stand on its edge but will never land on it).
Nevertheless, we can describe a contrived quasistatic regime that leads to that conclusion. Admittedly, dice are never cast this way but it's an idealized limitof a physical situation and fair dice ought to be fairunder any conditions, including those (like isohedra are). Therefore, fair dice must be equispherical (all faces must subtend the samesolid angle).
First, we assume that the horizontal plane on which dice land is infinitely sticky,so that a die can pivot about a vertex or an edge but will never roll. Thus, it can come to rest on a surface that couldn't serve as a stable resting placewithout such stickiness (aspindle could land on its pyramidalextremities).
Second, we assume dynamical (inertial) effects are negligible. This would happen if the die was moving at low speed in a very viscous fluid. We may also assume that this fluid is only slightly less dense than the homogeneousstuff the die is made from. The die is fully immersed in the fluid, at rest in a random orientation. Upon release, it will essentially fall at constant speed (its terminal velocity) without rotating.
discussed below is an idealized way dicecould actually be cast. What's crucial is the fact that this describesan actual physical situation (or, at least, the idealized limit of such situations). The conclusions derived from this contrived model would thus be valid for any diewhich is intrinsically fair (i.e., unconditionally fair) in the sameway convex isohedra are: A fair die lands with equal probability on any of itsfaces regardless on the surrounding conditions, provided it starts at rest in a randomorientation.
Under such contrived conditions, the die will simply land on whichever face is directlybelow its center of gravity.
The assumption that the die is initially randomly oriented means that the downward vertical through the center of gravity crosses aface with a probability proportional to thesolid angleit subtends, as seen from the center of gravity. That same probability is also the probability that the die will land on theprescribed face.
For example, anisogonal truncated octahedronmakes a fair die under such quasistatic consitions (dead-cat bounce) when the ratio of an edge between hexagons to the side of a square face is:
x = 0.32173356003298450750124...
(2013-02-18) The bottom face of a die tends to be closest to its center of gravity.
When placed on an agitated horizontal plate mimickingthermal motion, dice would naturallytend to orient themselves in the most energetically favorable way. That's achieved by lowering the center of gravity as much as possible.
By reducing the amplitude of the agitation gradually until motion freezes,we effectively cast the dice in a way that definitely favors,for the bottom position, the faces which are closest to the center of gravity.
(2013-04-07) A mesohedron is an equispherical orthohedron.
An equispherical die that's fair under the previously describedquasistatic conditions cannot be fair with the thermal tossing method as well, unless all its faces are also equally distant from the center of gravity.
Thus, all the faces of a fair die should be equally distant from thecenter of gravity and also subtend the same solid angle. Isohedra clearly meet both conditionsby symmetry. Below are examples ofnon-isohedral dice that satisfy this restriction (the solids that don't cannot possibly be fair dice). Let's establish the vocabulary:
An orthohedron isdefinedas a polyhedron with an inscribed sphere; the center of that sphere isequally distant from all the faces.
If all the faces of an orthohedron subtend the same solid angle (i.e., if it's equispherical ) then we call it a mesohedron.
If the mass distribution of a solid orthohedron is such that theaforementioned center isat the center of gravity, we say it's balanced.
The above shows that a fair die must be a balanced mesohedron.
A balanced isohedron with spherical inertia (i.e., its threeprincipal moments of inertia are equal) is a fair die under all tossing conditions, by reasons of symmetry. So is a balanced isohedron with an axis of symmetry and mere cylindrical intertia about that axis (the moment of inertia about the axis of symmetry candiffer from the moments about perpendicular axes). This later case applies to bipolar dice (amphihedra or deltohedra). Whichever relevant condition is automatically satisfied forisohedral solids of uniform mass density (ordinary unloaded dice).
By contrast, the fairness of a balanced mesohedron isonly guaranteed for the two extreme casting methods described above (quasistatic or thermalregimes) which we may loosely think of as dead cat bounce and high resilience, respectively. For intermediate conditions, no such guarantee exists.
(2013-04-07) Symmetrical mesohedron (equispherical orthohedron) with 10 faces.
The so-called fitness dice depicted at leftare a pair of 10-sided latex rubber dice meant to be rolled togetherto suggest a type of physical exercise anda number of repetitions to perform. The dice are large enough (7" height) to be tossed on a gym floor. A pair retails for about $30.
Those seem to be shaped roughly like 10-sided mesohedra. Let's describe what a perfect 10-sided mesohedron would be:
A die with that general appearance will be equispherical if and only if bothsquare faces subtends a solid angle of 2/5 each.
Indeed, by symmetry, the rest of the spat (4) is shared equally among the eight other faces, so each of those alsosubtends a solid angle of 2/5.
On the other hand, a polyhedron is orthohedraliff one point (the center) belongs to all bisectors of itsface angles (i.e., the dihedral angles formed when two faces meet at an edge).
A mesohedron, is an equispherical orthohedron. Putting both of the above conditions together yields the following cross-section (in a plane perpendicular to half of the horizontal edges). The angle is determined by the aforementionedequisphericity condition:
= Arctg 1/5¼ = 33.7722424...°
Dimensions of a Mesohedral Decahedron ( of height 5¼ )
The above area of a trapezoidal face is equal to its height /2 multiplied into the half-sum of its bases (1+)/2. It could also be obtained with Brahmagupta's formula (since an isosceles trapezoid is indeed a cyclic quadrilateral ). The total volume is twice the volume of a conical frustum.
Note one similarity with the geometry of a sphere: The surface area of this polyhedron happens to be four times the area of itsequatorial cross-section.
(2013-04-23) Making an orthohedral rhombic pyramid equispherical and balanced.
Consider a tetragonal pyramid with two vertical planes of symmetry. Its horizontal base is a rhombus (i.e., an equilateral quadrilateral) and it has an inscribed sphereby reasons of symmetry, since the bisectors at the four horizontal edgesdo intersect at a point on the vertical axis.
Such a shape depends on three parameters; the vertical height h and the half-diagonals of the horizontal rhombus, x and y. Alternately, we could consider as parameters the three differentside lengths, a, b and c :
a 2 = x 2 +h 2b 2 = y 2 +h 2c 2 = x 2 + y 2
One parameter determines the size. We may adjust the other two to makea uniform-density solid both equispherical and balanced. Here it goes:
The tangent of the dihedral angle based on an horizontal edge is hc/xy. (: xy/c is the altitude of the right triangle of sides x and y.) The inclination of the bissecting plane (with respect to the horizontal) is halfthe angle corresponding to that... Now, in a straight pyramid of uniform mass density, the center of gravity is at a height z = h/4. Thus, the tangent of the bissector's inclination must be ¼ which makes the original dihedral angle's tangent 2t/(1-t2) = 8/15. All told, the solid is balanced when:
hc / xy = 8 / 15 or z (x2+y2)½ / xy = 2 / 15
The solid is equispherical with respect to the center at altitude z if and only if the base subtends a solid angle of 4/5. Indeed, since the other 4 faces share equallythe rest of the spat (4) by symmetry,this make them subtend the same solid angle of 4/5. Using our expression for the solid angle subtended by a rhombus, this condition translates into:
(2013-04-07) Seven-sided mesohedron with a ternary axis of symmetry.
To obtain an heptahedron with a vertical ternary axis of symmetry,we may truncate off horizontally one pole of a bipolar polyhedron witha vertical ternary axis. That's to say, either a trigonal dipyramid or a trigonal deltohedron.
In either case, the final solid can't be an orthohedron unless the untruncated side is flatter than the side of the truncated pole.
(2013-02-09) The rôle of aspect ratio : From coin to rod.
A cylindrical die is an homogeneous dieallowed to bounce repeatedly on an horizontal plane, without loss of energy.
(2013-04-19) A simplified analysis...
Let's consider a polygonal wheel bouncing on an horizontal track.
(2012-11-28) 
(strombic icositetrahedron or trapezoidal icositetrahedron ). It's marketed by GameScience (Zocchi's company) under the name of D-Total, featuring fancy markingsthat are meant to facilitate the use of the die as a substitute for dice with 2, 3, 4, 5, 6, 7, 8, 10, 12, 20, 24, 30, 40, 50, 60, 70 or 80 sides. This is jointly credited to Dr. Alexander F. Simkin, Frank Dutrain (of LD Diffusion) and Louis Zocchi (2009).
(
(2013-03-27) 
= Arctg 1/5¼ = 33.7722424...°
