The Five Perfect Solids

Five shapes are perfectly regular. Every one of them is in your dice bag, and there will never be a sixth.

Why Only Five

Pick up a d20 and turn it over in your hand. Every face is the same triangle. Every edge meets its neighbour at the same angle. Every point has the same five triangles crowded around it. Put it down, pick it up again, and you cannot tell which way it faced before.

That is what a mathematician means by a regular solid: identical faces, identical edges and identical corners. Rotate the shape until any face sits where another one was, and nothing has visibly changed. Ask also that it be convex, no dents and no spikes turned inward, and you have described a very small club.

It has five members. Not five that happen to be useful, or five that survived into modern manufacture. Five that can exist at all. Euclid proves it in the last proposition of the Elements, Book XIII [1].

The reason is easier than it sounds. A solid corner needs at least three faces meeting at a point, and their angles must add up to less than a full turn (360°). Hit 360° exactly and the corner lies flat: a tiled floor, not a box.

The Corner Test

Faces at each corner	Angle sum	Gap left	Result
3 triangles (60° each)	180°	180°	tetrahedron, d4
4 triangles	240°	120°	octahedron, d8
5 triangles	300°	60°	icosahedron, d20
6 triangles	360°	0°	flat. No solid.
3 squares (90° each)	270°	90°	cube, d6
4 squares	360°	0°	flat
3 pentagons (108° each)	324°	36°	dodecahedron, d12
3 hexagons (120° each)	360°	0°	flat. A honeycomb.

Every other polygon has corners wider than 120°, so three of them already overshoot. The list runs out there. Not a limit of engineering or imagination: a limit of space.

What That Buys You

A die is fair when no face is favoured over another, and perfect regularity is a very thorough way to guarantee it. No heavier corner to settle onto, no broader face to prefer, no axis it would rather spin around. Symmetry does the work a manufacturer would otherwise do with tolerances and testing.

So five ancient shapes ended up in a plastic bag in your rucksack. Nobody designed them for gaming. They were the only options.

The Five at a Glance

"Roundness" here is sphericity: 1.0 is a perfect ball. It is computed from each solid's volume and surface area, and it tracks how well each one rolls.

Solid	Faces	Face shape	Roundness	At the table
Tetrahedron	4	triangle	0.67	d4
Hexahedron (cube)	6	square	0.81	d6
Octahedron	8	triangle	0.85	d8
Dodecahedron	12	pentagon	0.91	d12
Icosahedron	20	triangle	0.94	d20

The mathematics is usually credited to Theaetetus (c. 417–369 BCE). An ancient commentary gives him the octahedron and icosahedron and credits the cube, pyramid and dodecahedron to the Pythagoreans before him; Book XIII of Euclid rests on his work [2]. The four elements came from Empedocles. Plato welded the two together, and the join has held for two and a half thousand years.

Plato's Elements

In Timaeus, written around 360 BCE, Plato gives four of the solids to the four elements. He is careful about it. "In assigning this figure to earth," he says, "we adhere to probability" [3]: the likely account, argued from what the shapes are like, not handed down as fact. His reasoning runs on sharpness, size and how readily a thing moves.

Tetrahedron: Fire

Four triangular faces and the sharpest corners of the five. The solid with the fewest faces, Plato argues, "must necessarily be the most moveable, for it must be the acutest and most penetrating in every way" [3]. Fire.

It is also the worst roller in the bag. With four faces it barely tumbles, and it lands with a point upward, so you read the number at the tip or along the bottom edge, depending on the set. On the floor at two in the morning it is point upward too.

Hexahedron: Earth

The cube, the only one of the five with square faces. "To earth, then, let us assign the cubical form; for earth is the most immoveable of the four" [3]. The steadiest shape went to the steadiest element. It is also the oldest gaming die by a wide margin, and the only one most people outside this hobby have ever handled.

Octahedron: Air

Two square-based pyramids joined at the base. Between fire and water in size, sharpness and mobility, it took the element between them.

Look at one and it seems to have a single waist, a top half and a bottom half. It does not. The numbers are printed upright against one plane, inventing a horizon the shape has no opinion about. Turn it and you find three square cross-sections, not one.

Icosahedron: Water

Twenty triangles. Plato gives water "the greatest" body of the three and the least mobile [3], because water pours rather than darts. Two thousand years later it became the die every roll in the game passes through, and it is the roundest of all five (0.94 in the table above), which is why it rolls best.

Dodecahedron: the Universe

Twelve pentagons, and the odd one out. Plato builds his elements from two right triangles: half an equilateral triangle, and half a square. Fire, air and water are faced with the first, which lets them break down and rebuild into one another from shared stock. Earth, made of squares, cannot join in.

A pentagon comes from neither triangle, so the dodecahedron sits outside the scheme. Plato gives it one sentence: "There was yet a fifth combination which God used in the delineation of the universe" [3]. Later writers handed it aether.

Which makes the d12 the least-used die in the bag and, by some way, the one with the grandest job description.

The Ones That Are Not

Your bag holds more than five dice, and everything past the five is a compromise. They are not lesser dice. Most are perfectly fair. They buy their fairness with less symmetry than a Platonic solid has to spend.

The d10

Ten kite-shaped faces, five above the waist and five below. It is a pentagonal trapezohedron, and it is not regular: two sharp corners at top and bottom, where five kites meet, and ten blunter ones zigzagging round the middle, where three meet [4]. Slice off the two sharp tips and you get a regular dodecahedron [4], so the d10 is a d12 wearing a party hat on each end.

It is still perfectly fair, and that is worth sitting with, because it shows what fairness needs. As the physicist Greg Gbur puts it, "we really only need every face of the die to be equivalent" [5]. The Platonic solids have more symmetry than a die requires. The d10 has exactly enough.

Look closely and its faces do not line up across the middle; they are staggered. They must be. Line them up and, in Gbur's words, "the die would come to rest on one face, but an edge would be facing upward!" [5]

The d100

Lou Zocchi's Zocchihedron (1985): a ball about 1.5 inches across with 100 flattened spots, nicknamed "Zocchi's Golfball" [6]. It pays for the ambition. It rolls for an embarrassingly long time; later versions hid free-falling weights inside to make it settle sooner. And a White Dwarf test found the numbers crowded near the poles, making results above 93 or below 8 noticeably rarer than middling ones [6].

Most tables roll two d10s instead, one for tens and one for units. Faster, fairer, and a great deal less dramatic.

The d2 and the d3

There is no two-sided solid and no three-sided one, but real d2s and d3s exist. Gbur describes both: a d3 is a triangular tube with rounded edges, and a d2 is a cube-like body where three faces in a "U" read 1 and the other three read 2, rounded so each trio rolls as one side [5].

The cheap version needs no special die at all: read a d6 as 1–2 = 1, 3–4 = 2, 5–6 = 3 for a d3, or odd/even for a d2.

You could flip a coin. But you did not come here to flip coins.

Which Die Does What

Dice are small sometimes-pointy things made of plastic or metal (or sometimes wood or gemstone!) that go clickety-clack and then give you a number.

SwiftSign, D&D Beyond forums, 2020 [7]

That is most of it. In the current rules (SRD 5.2), the d20 decides whether a thing happens: ability checks, saving throws and attack rolls are all "D20 Tests" [8]. The other dice decide how much: damage, healing, hit points. The rules always name the die, so nobody must memorise this table. You will anyway, within three sessions.

Every Die and Its Job (SRD 5.2)

Die	Average	Rolled for	At the table
d20	10.5	every D20 Test	"roll to hit"
d12	6.5	greataxe; Barbarian Hit Point Die	"the barbarian's friend"
d10	5.5	heavy crossbow; Fighter, Paladin, Ranger hit dice	"tens and units"
d8	4.5	longsword, rapier, battleaxe; hit die for half the classes	"roll damage"
d6	3.5	shortsword, mace, Sneak Attack, fireball	"eight dee six, everyone"
d4	2.5	dagger, magic missile, Potion of Healing (2d4 + 2)	"ow"

Weapon dice and hit dice are from the SRD 5.2 weapon table and class traits [8]. Six of the twelve classes (Bard, Cleric, Druid, Monk, Rogue, Warlock) roll a d8 for hit points; Sorcerer and Wizard roll a d6, the only place a d6 beats a d8 at being sad.

Worked Examples

One Round, Three Dice

Your fighter swings a longsword at a Goblin Warrior (AC 15, 10 hit points) with +5 to hit and +3 Strength.

• Attack 1d20+5: you roll 14, total 19. Equal to or above 15: a hit.
• Damage 1d8+3: you roll 6, total 9 slashing.
• Result: the goblin has 1 hit point and a new outlook on life.

One swing kills it only on a 7 or 8 on the d8: a 25% chance. Bring a friend.

Your Odds to Hit

With +5 to hit, a natural 1 always misses and a natural 20 always hits [8]. Advantage means roll two d20s and keep the higher; Disadvantage, the lower [8].

Target AC	Normal	Advantage	Disadvantage
10	80%	96%	64%
12	70%	91%	49%
14	60%	84%	36%
15	55%	80%	30%
16	50%	75%	25%
18	40%	64%	16%
20	30%	51%	9%

Advantage is worth the most in the middle of the table, where it can add 25 percentage points. Against AC 10 you would have hit anyway.

The Famous Ones

• Fireball 8d6: average 28. Eight times in ten the total lands between 22 and 34. The maximum, 48, comes up about once in 1.7 million casts. The table leans in for the count every time regardless.
• Magic missile: three darts of 1d4+1, 10.5 on average, never fewer than 6. The caltrop's finest hour.
• Ability scores 4d6, drop the lowest, six times: this is the SRD's "Random Generation" method [8]. Average 12.24. You roll 15 or better 23% of the time, 8 or worse 10.5%, and a perfect 18 about once in 62 rolls.

Reading Percentile

Roll two d10s: one is the tens, one the units. A 4 on the tens and a 7 on the units is 47. Double zero is conventionally read as 100 [4]. This is why sets ship a second d10 marked 00–90: it ends the argument about which die was which.

The roller has every die from d2 to d100, and the 3D table has a drop-lowest button for ability scores.

A History of the Dice

Older Than Every Empire

Dice are older than the games we play with them by thousands of years. The Egyptian board game senet, played before 3000 BCE, moved its pieces by flat two-sided throwsticks. Before that, people cast the ankle bones of hoofed animals (knucklebones) to tell fortunes. Rome had two kinds of dice: large tali marked on four sides, and small cubic tesserae numbered one to six [9].

And twenty-siders are ancient. The Metropolitan Museum of Art holds a faience icosahedron from Egypt, dated 2nd century BCE to 4th century CE, with the first twenty letters of the Greek alphabet on its faces. The museum suggests dice like it may have picked which part of a public oracle to consult [10]. Your d20 has been telling people their fortune for two thousand years. It has only recently started telling them they fell in the pit.

How D&D Got Its Bag

In the early 1970s a d20 came from Japan or Britain, at a cost and a wait. Then, as the historian Jon Peterson traces it, an American source turned up: Creative Publications of California, which sold its twenty-sider only in a set with the other four Platonic solids [11]. That is why your bag holds exactly these shapes. Wargame publisher Don Lowry stocked the sets, and in June 1973 Gary Gygax published "Four & Twenty and What Lies Between," conceding that "the most useful are the 20-sided dice" [11].

The Crayon Years

Early d20s were numbered 0–9 twice. Gygax's fix: "you can color in one set of numbers on the die," and read the coloured half as ten higher [11]. Dice were a real bottleneck: in Peterson's words, "one could easily photocopy rules, but not dice" [11]. TSR resold the Creative Publications sets at a mark-up that climbed from $1.75 to $3.00, and only in the 1980s added a newcomer to the bag: the d10 [11].

A Timeline

Year	What happened	Source
1973	Gygax's "Four & Twenty" article; Creative Publications sets on sale through wargame mailers	[11]
1974	Dungeons & Dragons published; polyhedral dice quickly become its signature	[11]
1977	Holmes Basic Set ships with a set of polyhedral dice	[12]
1979	Dice shortage: some boxes ship cardboard number chits and a coupon instead	[12]
1981	Moldvay Basic Set: six polyhedral dice and a wax crayon to colour the numbers	[12]
1985	Zocchi's hundred-sided "golfball"	[6]
2025	SRD 5.2: every check, save and attack is a D20 Test, with Advantage and Disadvantage	[8]

Why We Roll Our Own

In 1967 the sociologist James Henslin studied cab drivers who shot craps, and recorded their belief that a hard throw produces a big number and a soft throw a small one [13]. Eight years later Ellen Langer named the wider effect the illusion of control: people feel more confident of a chance outcome when they choose, practise or act for themselves, though the odds do not move [14]. It is why a player reaches across the table rather than let the DM roll their save.

It changes nothing about the dice. It changes everything about how the roll feels.

Dice jail is not for the dice. It is for us.

Open a new set today and you get the five perfect solids plus two d10s: seven dice, five of them perfect, all of them older than the game they serve.

This post is also a free booklet: download the PDF.

Table Talk

Arthur: Five regular solids. Euclid proved it, and no dice company will ever sell you a sixth.

Tia: I care less about the maths than the player who puts a d20 in dice jail for rolling a 1.

Arthur: The d20 did nothing wrong. It is the roundest shape in the bag. Every face has the same chance.

Tia: Langer would say the player knows that. Jail is how we feel in control when the odds don't move.

Arthur: Then jail the player, not the die.

Tia: And the d10? It isn't even regular.

Arthur: Fair anyway. A die only needs every face to be equivalent. Perfect symmetry is overkill.

Tia: So the d12 is overqualified, underused, and holds the universe. Relatable.

Arthur: Practical tip: no d3 in the bag? Roll a d6 and read 1–2 as 1, 3–4 as 2, 5–6 as 3. No golfball needed either. Two d10s, tens and units.

References

1. Euclid. Elements, Book XIII, Proposition 18 and following remark (c. 300 BCE). Trans. D. E. Joyce, Clark University. https://mathcs.clarku.edu/~djoyce/java/elements/bookXIII/propXIII18.html
2. O'Connor, J. J., & Robertson, E. F. "Theaetetus of Athens." MacTutor History of Mathematics, University of St Andrews. https://mathshistory.st-andrews.ac.uk/Biographies/Theaetetus/
3. Plato. Timaeus (c. 360 BCE), 53c–56c. Trans. Benjamin Jowett. Project Gutenberg. https://www.gutenberg.org/ebooks/1572
4. "Pentagonal trapezohedron." Wikipedia (geometry and d10 use). https://en.wikipedia.org/wiki/Pentagonal_trapezohedron
5. Gbur, G. (2017). "The geometry of weird-shaped dice." Skulls in the Stars. https://skullsinthestars.com/2017/03/09/the-geometry-of-weird-shaped-dice/
6. "Zocchihedron." Wikipedia (citing White Dwarf testing). https://en.wikipedia.org/wiki/Zocchihedron
7. SwiftSign (18 Aug 2020). "What do the dice do???" D&D Beyond Forums, Tips & Tactics. https://www.dndbeyond.com/forums/dungeons-dragons-discussion/tips-tactics/78441-what-do-the-dice-do
8. Wizards of the Coast (2025). System Reference Document 5.2.1, "D20 Tests", "Rolling 20 or 1", "Advantage/Disadvantage", "Generate Your Scores", class Core Traits tables, Weapons table. CC-BY-4.0. https://www.dndbeyond.com/srd
9. "Dice." Wikipedia (history section). https://en.wikipedia.org/wiki/Dice
10. The Metropolitan Museum of Art. "Twenty-sided die (icosahedron) with faces inscribed with Greek letters," Ptolemaic–Roman Period, 2nd c. BCE–4th c. CE. https://www.metmuseum.org/art/collection/search/551072
11. Peterson, J. (3 Feb 2013). "How Gaming Got Its Dice." Playing at the World. http://playingattheworld.blogspot.com/2013/02/how-gaming-got-its-dice.html
12. "Dungeons & Dragons Basic Set." Wikipedia. https://en.wikipedia.org/wiki/Dungeons_%26_Dragons_Basic_Set
13. Henslin, J. M. (1967). Craps and magic. American Journal of Sociology, 73(3), 316–330. https://doi.org/10.1086/224479
14. Langer, E. J. (1975). The illusion of control. Journal of Personality and Social Psychology, 32(2), 311–328. https://doi.org/10.1037/0022-3514.32.2.311