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Final Answers
© 2000-2018   Gérard P. Michon, Ph.D.

  Remond de Montmort  Blaise Pascal


God doesn't play  dice  with the Universe.
Albert Einstein   (about  quantum mechanics)
Formerly on this page:
  • Two chiral fair dice:  No mirror symmetry,  24  or  60  pentagonal faces.
  • A pseudo-isohedral die.  Its faces are congruent, but is it fair?
    The above articles have moved...    Click for the new location.
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 Pentagonal Hexecontahedron

Related articles on this site:

Related Links (Outside this Site)

Unfair Dice  by  Ivars Peterson  (MAA, 1998-10-26).
Fair Dice  by  Ed Pegg, Jr.   (Math Games, MAA, 2005-05-16).
Doll Dem' Bones:  From astragali to dice, making sense of randomness.
The design of asymmetric dice  by  Michael Lugo  in  "God Plays Dice".
Fair but irregular polyhedral dice  by  Joseph O'Rourke  in  MathOverflow.
What kinds of fair dice?  by  Ipetrich & al.  in  "Rational Skepticism".
Dice rolls are not completely random  by  Ben P. Stein  (2012-09-12).
Awsome Dice Blog's epic 20,000-roll dice randomness test  (Sept. 2012).
N nodes "evenly" arranged on a sphere, using the golden ratio  (Aug. 2004).
Dice of N dimensions  by  Jonathan Bowers.
Vauchier-Playbox  (France, 1920).
Alea Kybos' Dice CollectionShapes.
Polyhedral Dice  by  George W. Hart (1999).
Ancient Dice  by  "The Cartographer"  Al-Zahr (i.e., "the Dice").
What shapes do dice have?  by  Kevin Cook  (largest collection of dice).
Introducing D14, D18  (and D22)  spherical dice:  A KickStarter project.
Mathematician's Dice  by  by Matt Chisholm,  A KickStarter project.
Grand Illusions, Ltd. :   Math toys  (for sale).
Tannen's Magic Store :   Dice magic  (for sale).
Robert Fathauer's MathArtFun :   Polyhedral dice  (for sale).
The Dice Shop  |  Gamestation  |  Dark Elf Dice  |  Great Hall Games
Mini Dice Towers  |  Gaffed Dice

Wikipedia :   Dice  |  Isohedra  |  Knucklebones

Numerically Balanced Dice  by  Robert Bosch   (G4G Celebration, 2016).
New Polyhedral Dice  by  Robert Fathauer   (G4G Celebration, 2016).
 Backgammon precision dice


(2012-11-26)   Standard spots on cubic dice :
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 the dices in the two packs were  not  alike because of the arrangement of the spots.

 Come back later, we're
 still working on this one...

(2012-11-29)   Unrestricted Dice
Let's drop the rule that requires opposite faces to add up to  7.

In this section, a "configuration" is understood to be merely a labeling of the faces of the die, regardless of the orientation of the labels...  To each such configuration correspond  8  different orientations of traditional markings with spots.  Thus, the  30  configurations enumerated below correspond to  240  ways to manufacture a die.

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 opposing each 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 further choices for setting the remaining two faces  (that are opposite to each other).  That's  12  configurations to add to the  4  enumerated in the previous paragraph, for a grand total of  16.

All told, in the 30 ways to number a dice, there are two ways  (right-handed and left-handed)  in which opposing sides always add up to seven and  16  ways in which they never do.

(2013-02-06)   A 9-hedron stable on only 3 faces   (3-spindle)
An  elongated n-gonal dipyramid  is a substitute for a  fair  n-sided die.
 Elongated Triangular Dipyramid   If its pyramidal components aren't too flat, such a 3n-hedron is only stable on an horizontal plane if its rests on one of its  n  lateral faces.  If the die has an order-n axis of symmetry, all those faces are equiprobable because they're equivalent  (assuming uniform mass density).

A nicer and more elaborate solution :   Three-sided dice  by  Don Simpson.
Dice alternatives and substitutes
Monostatic polytopes  are "dice" with a single stable facet...

 Hellerstein's nontransitive set of 3-sided dice
(2012-11-28)   Nontransitive set of dice.
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 HellersteinCCSF :

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:

   3    5    7  
  2   Yes Yes Yes
4  Yes Yes
   1    6    8  
  3    Yes Yes
5  Yes Yes
7   Yes
   2    4    9  
  1   Yes Yes Yes
6   Yes
8   Yes

Plus Magazine :   Curious Dice  by  James Grime.
Winning Odds  by  Yutaka Nishiyama and Steve Humble.
Nontransitive dice for sale  (Grand Illusions) :   4 Dice   |   3 Dice

(2012-11-28)   Sicherman's Pair of Dice
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 involves generating 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 several dice is the coefficient of  xn  in the product of their polynomials  (HINT:  to obtain that term, you must sum up all the ways there are to pick one term from each factor so 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 coefficient of  xn  in the  square  of the above polynomial.  Namely:

S2   =   x2 + 2x3 + 3x4 + 4x5 + 5x6 + 6x7 + 5x8 + 4x9 + 3x10 + 2x11 + x12

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.  QED

What's somewhat miraculous is that we end up with a pair of  6-sided  dice.  To match what's done with traditional dice, those dice should be built with 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.

Wikipedia   |   Grand Illusions   |   Col. Sicherman's home page (CV)
"Sicherman dice" by   Ed Pegg JrAlexander BogomolnyArne Ledet,  ...
Plus Magazine :   Let 'em Roll  by  Clare Hobbs  (2006).

(2013-02-05)   Rolling several dice instead of one.
When does the sum of two dice give equiprobable totals?

The standard set of seven polyhedral dice made popular by  Dungeons & Dragons consists of the five platonic solids and a pair of 10-sided pentagonal deltohedra.  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.

This exception wouldn't be needed if the low die was marked  1-10  instead of  0-9  (all other standard dice do start with 1).  Some decimal dice are available which allow just that; they're simply not popular.

More generally, we may consider a set of  p  n-sided fair dice where the  j+1st  face of the  i+1st  die is marked j.n).  When those dice are rolled, they give any total from  0  to  np-1  with equal probability.  Let's generalize:

For a prescribed integer  M,  what are the sets fair dice marked integers that will give any total between 0 and M-1 with probability 1/M ?

Well, the polynomial approach introduced in the previous section reduces 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 first non-trivial factorization is for  M = 6:

1 + x + x2 + x3 + x4 + x6   =   (1+x) (1-x+x2 ) (1+x+x2 )   =   (1+x3 ) (1+x+x2 )

A factorization gives a legitimate set of dice only if all the factors are polynomials whose coefficients 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)   Polyhedral Dice
Dice people actually use, for divination or recreation.

Rôle playing games  (RPG)  call for a variety of dice besides the traditional 6-sided cubic dice  (D6).  The most popular sets have  7  dice:  7 polyhedral dice

  • A regular tetrahedron  (D4).
  • A cube  (D6).
  • A regular octahedron  (D8).
  • A regular dodecahedron  (D12).
  • A regular icosahedron  (D20).
  • Two pentagonal deltahedra  (D10)  used as percentile dice  (D100).

 Ancient glass icosahedron 
 (blue, severely chipped)  Icosahedra were already used in Antiquity, for divination purposes.  The large  (52 mm)  glass die shown 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 and was expected to fetch between $4000 and $6000.  It would be worth a lot more now.

 3 Astro-Dice   For 12-sided dice, the  regular dodecahedron  has eclipsed the  rhombic dodecahedron,  which was apparently mass-produced only once, around 1978, for the  Ask Astro-Dice  fortune-telling game.
For the shot at left,  I got an old set from  eBay  ($15 on 2013-03-29).  New ones use regular dodecahedra, unfortunately...

Note that Pluto  (top symbol on the center die)  was still a planet back then.

 Tetrakis Cube  In recent years, two distinct isohedra with  24  faces have 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   Deltoidal Icositetrahedron (strombic icositetrahedron  or  trapezoidal icositetrahedron ).  It's marketed by  GameScience  (Zocchi's company)  under the name of  D-Total,  featuring fancy markings that 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).

GameScience Sales Pitch,  by  "Colonel" Lou Zocchi :  1  |  2  (2008-08-17)

(2013-04-26)   Commercial Dice Sizes
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)
12 mm
15 mm
19 mm
28 mm
35 mm
D4 17.4 mm   
D612 mm15.0 mm19 mm25 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 mm24 mm  
D30   32 mm 

Oversized dice could damage dice trays.  They're best tossed on carpets.

 Click Here 
 for Details (2013-04-14)   Convex Isohedra are  fair dice.
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.

The dual of an isohedron is an isogonal polyhedron, and vice-versa  (duality  being understood with 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  (polyhedral or 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 13 Archimedean 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).  Isogonal Tetrahedron

For example, the familiar regular tetrahedron is  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  chiral  iff  the three quantities  a, b  and  c  are distinct.

The full classification of  all  isohedra is given  elsewhere on this site.

 Round Dice
 (2013-02-22)  The Secret of Round Dice
A steel ball moves in an  isogonal  cavity,
dual  of the isohedral outer pattern.

Currently,  "6-sided"  spherical  dice are available  (the inner cavity is  octahedral ).

Any  isohedral  die could be made this way, by carving out a cavity in 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 cavity was 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 where the outer markings would  seem  asymmetrically distributed.  Yet, it would be a perfectly fair die and, therefore, a great conversation piece...

(2013-02-14)   Introducing the  scalene  isogonal tetradecahedron :
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, the isogonal polyhedron is said to be  scalene.

On the other hand, an isogonal polyhedron where at least two adjoining edges have the same length can be called  isosceles.  Any polyhedron where all edges have the same length is said to be  equilateral.  A polyhedron that's both isogonal and equilateral is said to be  uniform  There are  75 or 76 nonprismatic uniform polyhedra  (the convex ones are the 5 Platonic solids and the 13 Archimedean solids).

Consider now the three distinct types of  isogonal tetradecahedra  obtained by cutting off the eight corners of a cube with increasing severity:

 Cube  Truncated 
 Cube  Cuboctahedron  Truncated 

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 persists when  different  truncations are used for nonadjacent vertices of the cube.  The resulting  scalene isogonal tetradecahedron  is a zonohedron  featuring 3 edge lengths, 6 rectangular faces and  two  distinct types of hexagonal faces.  At right is my own  old school  cardboard model with 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 patterns are  inner  ones.)    Scalene isogonal tetrahedron

If all such truncations are alike, then the resulting isosceles solid has square faces and is more readily obtained by "classical" truncation of a regular octahedron, without creation of new edges  (the scalene version can't be so constructed, because the  8  planes supporting the hexagonal faces don't form an octahedron).  Therefore, the locution  isogonal truncated octahedron  can only denote the tetradecahedron with square faces  (and congruent hexagonal faces)  discussed next as the shape of traditional Korean dice.

By definition, an  isohedron  is said to be scalene if its dual is  (the dual of an isohedron is an isogonal polyhedron and vice-versa).  For example, the dual of the polyhedron just described is a scalene isohedron  (a fair die)  with 24 faces congruent to the same triangle  (scalene tetrakis hexahedron).

 Juryeonggu (Gerard Villarreal, TX. 2013-02-05)   Juryeonggu
What polyhedron is this  14-sided  Korean die ?

Uniform Truncated Octahedron

The  juryeonggu  is precisely an  isogonal truncated octahedron.

Because its tetragonal faces are square, it's a special case of the above solid.  The shape is fully specified by the 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 is another uniform tetradecahedron, as is the truncated cube.)  This Archimedean solid  can tile space without voids  (a non-uniform  juryeonggu  can't).  It's the basis for the near-optimal foam described by Lord Kelvin in 1887  (Kelvin's cell has the same vertices as the tetradecahedron but all its edges are curved and so are its hexagonal faces).

For any x, a  juryeonggu  has the following dimensions  (derived below).  The locution  "radius to"  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 NameValue
Side of a square face   (basic unit)a41
Length of an edge between hexagonal facesa6x
Diagonal of a square face Ö2
Diagonal of an hexagonal face 1 + x
Width of an hexagon  (between parallel sides) Ö3/2 ( 1 + x )
Radius of the circumscribed sphereR( 1 + x + ½ x2 )½
Radius of a square facer4Ö2/2
Radius of an hexagonal facer6Ö3/3 ( 1 + x + x2 )½
Radius to [center of] a square faceh4Ö2/2 ( 1 + x )
Radius to [center of] an hexagonal faceh6Ö6/3 ( 1 + ½ x )
Caliper ratio  ( h4 / h6 ) Ö3 (1+x) / (2+x)
Radius to [centers of] sides of squaresR4½ ( 3 + 4x + 2x2 )½
Radius to [centers of] other edgesR61 + x/2
Solid angle of a square faceW44 Arcsin [ 1 / (3+4x+2x2 )]
Solid angle of an hexagonal faceW6p/2 - 3 Arcsin (3+4x+2x2 )-1
Surface area of a square faceA41
Surface area of an hexagonal faceA6Ö3/4 (1 + 4x + x2 )
Total surface area  ( 6A4+8A6 )A6 + 2Ö3 (1+4x+x2 )
Volume of a square pyramid  ( A4 h4 / 3)V4Ö2/6 ( 1 + x )
Volume of an hexagonal pyramid  ( A6 h6 / 3)V6Ö2/24 (1+4x+x2)(2+x)
Total VolumeV6 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 either the diagonal of a square face  (length Ö2)  or an edge of length x.  In that equatorial plane, we also find  h4  and  R:

R2   =   [ 1 + (1+x)2 ] / 2   =   1 + x + ½ x2
h4   =   ( 1 + x ) / Ö2
R6   =   1 + x/2

An hexagonal face is obtained by subtracting three equilateral triangles of side x from an equilateral triangle of side 1+2x, so its surface area is:

A6   =   Ö3 / 4 [ (1+2x)2 - 3x2 ]   =   Ö3 / 4 ( 1 + 4x + x2 )

The diagonal of an isogonal hexagon with sides 1 and x is  (1+x).  The width between parallel sides is equal to that diagonal multiplied by the sine of  60°.  The least obvious quantity is  r6  which can be obtained from planar 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 established elsewhere on this site.  For hexagonal faces, we just use the fact that the solid angles subtended by all faces add up to  4p.

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   =   Ö( 3+4/Ö3 ) - 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 the inscribed sphere, by analogy with the isohedral concept).  This entails a quadratic equation whose positive solution is:

x   =   (Ö3 - 1) / 2   =   0.36602540378443864676372317...

In either case, it was guessed that endowing an  isogonal truncated octahedron  with a particular property that isohedra possess would endow them with the same fairness as isohedral dice.  It ain't quite so...

Oversimplifying the conclusions of the discussion below, the latter guess turns out to correspond to the situation where 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 a sticky surface).  Such a die would simply land on any face with a probability proportional to the solid angle  subtended by that face from the center of the polyhedron.  All faces subtend the same solid angle  (2p/7)  when:

x   =   (Ö((1 / 2sin p/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 between the two  (contrived)  extremes described above, a  juryeonggu  that looks fair in practice would have to correspond to a parameter  x  determined empirically under those 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 the following value of  x:

x   =     (N2 - 400) x1  +  (400 - N1 ) x2
N2 - N1

The two most interesting dice to build are the ones mentioned above:

x1   =   0.32173356003298450750124...
x2   =   0.36602540378443864676372...

If you build a third die using the above interpolation, you may roll your three dice many 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 or a 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).

 Carved Tetradecahedron  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 runsTotalPer face
Hexagon 7272767077708064 581s =
72.6 (16)
Square 2828243023302036 21936.5 (21)

s 2   =   A.B / (A+B)

Casting the 14-sided juryeonggu   Official blog of   Gyeongju,  South Korea   (2011-07-06)
juryeonggu.net   |   Juryeonggu images from Korean pages
"Tetrakaideccahedronnocube" "TET" & "Bocralette", marketed as Rolla-Strike (1985).
Justin Mitchell's (D14) Fourteen-Sided Dice.

(2013-03-27)   Quasi-fair and blatantly unfair dice
Discussing non-isohedral commercially available dice.

"Colonel"  Lou Zocchi  first earned a solid reputation in the manufacture and sale of  isohedral dice which he pioneered in 1974, riding the wave of the increasing popularity of  role playing games  (RPG).

For lack of symmetry among their faces, the fairness of non-isohedral dice may depend critically on the way they are cast and/or on the resilience of 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)  which I 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 by trial and error and tested extensively.  Sometimes, the necessary rigor isn't exercised in commercial endeavors...

 D5   The worst mass-produced offender is probably the 5-sided die  (D5)  pictured at left, a flawed design for which  Lou Zocchi  was granted US Patent 6926275  in 2005.

In his patent application, Zocchi stated that the device had been tested by rolling it 10163 times on plexiglass and 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 on one 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 cube until the outcome isn't  6  ("casino" dice are  machined  from extruded cellulose acetate to a precision better than 0.0005").  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-sided spindle or a 10-sided isohedron with duplicate labels.

 Blue Zocchihedron The celebrated  Zocchihedron  pictured at left is an interesting randomizing 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 come to a full stop but also makes it impossible to grasp its true dynamical state from its outer appearance.
US Patent D303553 (1989)   |   US Patent 6926276 (2005) 

Louis J. Zocchi (c.1935-) is also a part-time magician (excerpts: 1 | 2 )
Experimentally obtained statistics of dice rolls (6th Experimental Chaos Conference, Potsdam 2001).

(2013-01-27)   The Fair Dice Problem
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 on any face.

sufficient  condition for an homogeneous convex polyhedron to be 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 isometric transformation of the entire polyhedron  (i.e., a rotation or a mirror reflection mapping the polyhedron onto itself). 

An isohedron is thus a polyhedron where all faces are congruent to each other.  The converse need not be true.  For example, the convex deltahedra with 12, 14 or 16 faces aren't isohedral.

Isohedral symmetry is precisely what guarantees that all faces are 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 homogeneous rigid isohedral die are equiprobable, then it will certainly come to rest on any on its faces equiprobably.

This conclusion certainly holds for the Newtonian mechanics of  rigid bodies,  which will be our  only  concern in the rest of the discussion.  It would also hold for other mechanical laws, including special and general relativity, that can deal with the fiction of homogeneous and isotropic matter and are insensitive to chirality  (this last restriction comes from the fact that chirality-changing transformations are allowed as isohedral transformations).  The symmetry argument is otherwise general enough to deal with fragile isohedra made from (amorphous) gelatin or rubber.  Whenever the final integrity of such an isohedral die is sufficient to identify it as resting on some face, it will be any face with equal probability!

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 the correct shape using linear interpolation)  was based on the implicit assumption that dice are always cast the same way  (on an horizontal table covered with a particular shock-absorbing material, say).

Video :   Statistical Mechanics:  Lecture 1  (121 min)  by  Leonard Susskind   (2009-03-30)

(2013-02-07)   Necessary conditions for polyhedral dice to be fair
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 his Master's degree at UCCS was entitled  A Complete List of Fair Dice.  In it, the famous recreational mathematician actually classified  isohedra.  He was fully aware of the remote possibility that some  fair dice  might exist besides isohedra  (which are fair by symmetry)  but he clearly estimated  (rightly so)  that the less technical term was more suitable for a title.  Elsewhere, he answered the question  "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 standard conditions.  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, an intermediate height must exist 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 the aforementioned square pyramid can only be established if we analyze theoretically at least two sets of physical conditions.  The trick is to consider two limiting cases rather than any realistic motion.  A fair die could theoretically be thrown in any possible and couldn't show a difference in probabilities between those two limits, which turn out to be simple enough to analyze.

 Come back later, we're
 still working on this one...

   David Singmaster
David Singmaster
(2013-02-07)   Quasistatic Dice Casting
Slow  dead-cat bounce  on a sticky surface.

In 1981, David Singmaster  (b. 1939)  discussed the proposal that an homogeneous die would land on a face with a probability proportional to the solid angle subtended by 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  (whenever the 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 limit of a physical situation and  fair dice  ought to be fair under any conditions, including those  (like  isohedra  are).  Therefore, fair dice must be  equispherical  (all faces must subtend the same solid 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 place without such stickiness  (a spindle could land on its pyramidal extremities).

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 homogeneous stuff 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 dice could actually be cast.  What's crucial is the fact that this describes an 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 die which is intrinsically fair  (i.e., unconditionally fair)  in the same way convex isohedra are:  A fair die lands with equal probability on any of its faces regardless on the surrounding conditions, provided it starts at rest in a random orientation.

Under such contrived conditions, the die will simply land on whichever face is directly below its center of gravity.

The assumption that the die is initially  randomly oriented  means that the downward vertical through the center of gravity crosses a face with a probability proportional to the solid angle it subtends, as seen from the center of gravity.  That same probability is also the probability that the die will land on the prescribed face.

For example, an isogonal truncated octahedron makes 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)   Thermal Tossing of Dice
The bottom face of a die tends to be closest to its center of gravity.

When placed on an agitated horizontal plate mimicking thermal motion, dice would naturally tend 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.

 Gerard Michon (2013-04-07)   Mesohedron   (Michon, 2013)
A mesohedron is an  equispherical  orthohedron.

An  equispherical  die that's fair under the previously described quasistatic 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 the center of gravity and also subtend the same solid angle.  Isohedra  clearly meet both conditions by symmetry.  Below  are examples of non-isohedral dice that satisfy this restriction  (the solids that don't cannot possibly be fair dice).  Let's establish the vocabulary:

An  orthohedron  is defined as a polyhedron with an inscribed sphere; the center of that sphere is equally 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 the aforementioned center is at the center of gravity, we say it's  balanced.
  • The above shows that a fair die must be a  balanced mesohedron.

balanced isohedron  with spherical inertia  (i.e., its three principal 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 can differ from the moments about perpendicular axes).  This later case applies to  bipolar dice  (amphihedra  or  deltohedra).  Whichever relevant condition is automatically satisfied for isohedral solids of uniform mass density  (ordinary  unloaded  dice).

By contrast, the fairness of a  balanced mesohedron  is only guaranteed for the two extreme casting methods described above  (quasistatic  or  thermal regimes)  which we may loosely think of as  dead cat bounce  and  high resilience, respectively.  For intermediate conditions, no such guarantee exists.

Wikipedia :   Bisection

(2013-04-07)   Symmetrical Mesodecahedron   ( M10 )
Symmetrical mesohedron  (equispherical orthohedron)  with 10 faces.

 Pair of 10-sided rubber dice The so-called  fitness dice  depicted at left are a pair of 10-sided latex rubber dice meant to be rolled together to suggest a type of physical exercise and a 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 both square faces subtends a solid angle of  2p/5  each.

Indeed, by symmetry, the rest of the  spat  (4p)  is shared equally among the eight other faces, so each of those also subtends a solid angle of  2p/5.

On the other hand, a polyhedron is  orthohedral  iff one point  (the center)  belongs to all  bisectors  of its face angles  (i.e., the  dihedral angles formed when two faces meet at an edge).

Technically, a  bisector  is a set of two orthogonal planes consisting of all points equally distant from two intersecting planes.  In the case of a convex polyhedron, we may focus our attention to only a quarter of that figure  (a half-plane bordered by an edge of the polyhedron and featuring a nonempty intersection with its interior).

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  q  is determined by the aforementioned equisphericity condition:

 Hexahedral cross-section of 10-sided mesohedral die q   =   Arctg 1/5¼   =   33.7722424...°

Dimensions of a Mesohedral Decahedron   ( of height  5¼ )
Solid angle subtended by  any  face 2p/5 
Side of a square face  (circumpolar)1   1
Height of trapezoidal face (IJ=OJ)f/2   0.8090169943749474241...
Length of a slanted edge  (meridian)Ö3/2   0.86602540378443864676...
Length of an equatorial edgef   1.6180339887498948482...
Between  any  pair of parallel faces5¼   1.4953487812212205419...
Between antipodal polar edges  (2 OI)Ö(2f)   1.79890743994786727226...
Diameter  (largest width)f Ö2   2.2882456112707371904...
Surface area of a square face1   1
Surface area of a trapezoidal face¼+½f   1.0590169943749474241...
Total surface area4(1+f)  10.4721359549995793928...
Volume   =   5¼ (1+f) 2/3   2.60991595618192391525...

The above area of a trapezoidal face is equal to its height  f/2  multiplied into the half-sum of its bases  (1+f)/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 its equatorial cross-section.

 Blue Japanese D10 Trinity   The novelty item at left is a Japanese  Trinity  die  (as pointed out by Alea Kybos and Arjan Verweij).  It's a lesser alternative to the  isohedral  decahedron normally used by gamers  (a pentagonal deltohedron whose aspect was patented in 1983).

Wikipedia :   Bisection

(2013-04-23)   Balanced Mesopentahedron   ( M5 )
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 sphere by reasons of symmetry, since the bisectors at the four horizontal edges do 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 different side lengths,  ab  and  c :

a 2  =  x 2 + h 2             b 2  =  y 2 + h 2             c 2  =  x 2 + y 2

One parameter determines the size.  We may adjust the other two to make a uniform-density solid both  equispherical  and  balanced.  Here it goes:

The tangent of the dihedral angle based on an horizontal edge is  hc/xy.  (HINT:  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 half the 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  4p/5.  Indeed, since the other  4  faces share equally the rest of the  spat  (4p)  by symmetry, this make them subtend the same solid angle of  4p/5.  Using our expression for the  solid angle subtended by a rhombus,  this condition translates into:

1+Ö5     =     z (x2+z2 )½  +  z (y2+z2 )½
Vinculum Vinculum
4 (x2+z2 )½ (y2+z2 )½  +  z2

 Come back later, we're
 still working on this one...

Dimensions of the Balanced Mesopentahedron  ( M5 )
Solid angle subtended by  any  face 4p/5 
Inradius1   1
Height of the pyramid4   4
Dihedral angle between base and lateral faceArctg ( 8 / 15 )  28.0724869...°
Dihedral angle between adjacent lateral faces   
Dihedral angle between opposing lateral faces   

Solid angle subtended by a rhombus

(2013-04-07)   Balanced Mesoheptahedron   ( M7 )
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  with a 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.

 Come back later, we're
 still working on this one...

"Chestahedron" (2000)   by scultptor  Frank Chester  (b. 1939).

(2013-02-09)   On the Statistics of Bouncing Dice
The rôle of aspect ratio :  From coin to rod.

cylindrical die  is an homogeneous die allowed to bounce repeatedly on an horizontal plane, without loss of energy.

 Come back later, we're
 still working on this one...

"Physics of Dice"  by  Antonio Recuenco-Munoz   (May 2006).
"Unlikely Landings: Dice, Coins and the Mars Pathfinder"   by  Gary White.
"Dice Landing Probabilities"   by  Gary White  & al.   (Society of Physics Students)
"Predicting a Die Throw   (Science Daily, 2012-09-12)  from an AIP press release for:
"The three-dimensional dynamics of the die throw"   in  Chaos, 22, 8 pages  (December 2012)
by  Marcin Kapitaniak, Jaroslaw Strzalko, Juliusz Grabski & Tomasz Kapitaniak  (Lodz, Poland)

(2013-04-19)   Rolling Two-Dimensional Dice
A simplified analysis...

Let's consider a polygonal wheel bouncing on an horizontal track.

 Come back later, we're
 still working on this one...

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