d20 Dice Shape vs. Perceived Performance

I hope everyone has seen the videos of Lou Zocchi discussing dice. In the videos he argues (quite well) that typical dice are manufactured in such a way that causes them to be misshapen causing both their overall shape to not be perfectly round and their edges to be improperly rounded. He (and others) have suggested that this misshaping causes the dice to be like an egg in that the die will more often roll with the sides of the “egg” displaying rather than the poles. So I decided to do some tests of my own to see if there was any weight to this idea.

I have often seen individuals do dice rolling studies and come to the conclusion that the distribution of rolls is fairly even despite the perception that a die doesn’t roll well. I’ve seen larger studies with rolls in the thousands giving more complex conclusions. As I’m not looking to roll dice thousands of times to confirm distributions, I decided to look into the misshapen “egg” concept of my 7 d20s. Is my perception of the dice (good/cursed) have some connection to their overall shape? If so, that would be suggestive that there is something to the idea that the shape of the die affect die results.

In order to calculate the shape of my d20s I used a Neiko caliper (01407A) which has a resolution of 0.01 mm an accuracy of 0.02 mm. I used the axis length of each side measuring from the center of the face then used standard deviation to see how far each die varied from the mean of all axes. I always counted the axis going through the 20 face as axis 1 and then going clockwise then outward to identify the other axes:

Summary

I have 7 d20s and each I have obtained at different times, and each I have found performs well or more often than not seems to roll low numbers.

Die	Year Mfr.	Perception	Standard Deviation
Green	mid-80s	Good	0.029
Black	2010	Never Used	0.069
Orange Swirl	2009	Good	0.084
Blue-standard	1990s	Cursed	0.184
Blue-large numbers	1990s	Cursed	0.229
Pink	1990s	Rarely Used	0.280
Blue-bad paint	1990s	Cursed	0.333

Right away, there seems to be some credence to the idea of dice shape given that those with the lowest standard deviations seem to perform better. It also indicates that when the dice were manufactured might matter as the dice from the 1990s all have broad standard deviations.

Green

Obtained in what I believe was a BD&D Red Box in the mid 80s, it was one of those dice which came with the number unpainted (one used a provided white crayon for the user to fill in the numbers) and sported very sharp edges on each face.

Green	mm	axis	diff from mean
20-4	17.43	1	0.003
13-8	17.41	2	-0.017
14-10	17.44	3	0.013
9-6	17.49	4	0.063
16-3	17.41	5	-0.017
19-7	17.43	6	0.003
18-17	17.38	7	-0.047
15-5	17.41	8	-0.017
11-1	17.44	9	0.013
12-2	17.43	10	0.003
SD	0.0286937856

What’s interesting about this die is that the number distribution is different from modern d20s. Notice, for example, that the opposite number from the 20 is a 4 (not a 1) and the opposite number from the 19 is a 7 (not a 2). There is very little overall deviation from the mean axis length on this die with the deviations being in the hundredths of millimeters.

Black

This die came from a 4E Essentials Red Box and was manufactured with what seems like modern techniques, having painted numbers and rounded edges. Despite that, this die’s overall shape is comparable to the Green in that there is very little deviation from the mean axis length.

Black	mm	axis	diff from mean
20-1	18.93	1	-0.088
14-7	19.07	2	0.052
13-8	19.13	3	0.112
19-2	18.93	4	-0.088
17-4	18.99	5	-0.028
15-6	19.04	6	0.022
16-5	19.04	7	0.022
11-10	19.1	8	0.082
12-9	18.97	9	-0.048
18-3	18.98	10	-0.038
SD	0.0690893142

Unlike the Green, the number distribution here is using the now-common modern number distribution. For example, the opposite number from the 20 is a 1 and the opposite number from the 19 is a 2. The variation is slightly higher on axes 2, 3, and 8 but overall still quite tolerable. I’ve never used this die, but with the shape being quite good I intend to use it to see how well it seems to do in game.

Orange Swirl

This die was from a set purchased in 2009, and like Black the number distribution is modern and its shape has little deviation from the mean.

Orange Swirl	mm	axis	diff from mean
20-1	19.89	1	0.02
14-7	19.84	2	-0.03
13-8	19.95	3	0.08
19-2	19.75	4	-0.12
17-4	19.79	5	-0.08
15-6	19.97	6	0.1
16-5	19.96	7	0.09
11-10	19.94	8	0.07
12-9	19.85	9	-0.02
18-3	19.76	10	-0.11
SD	0.0843274043

I use this die often and it has always done well for me in that it rolls a good range of numbers.

Blue-standard

Now we start getting into the dice I obtained throughout the 1990s. This particular die is one of three blue die so I’ve named them based on the subtle characteristics of each.

Blue-standard	mm	axis	diff from mean
20-1	20.11	1	0.334
14-7	19.9	2	0.124
13-8	19.93	3	0.154
19-2	19.94	4	0.164
17-4	19.6	5	-0.176
15-6	19.57	6	-0.206
16-5	19.6	7	-0.176
11-10	19.71	8	-0.066
12-9	19.76	9	-0.016
18-3	19.64	10	-0.136
SD	0.1841014213

In this die one can see that the standard deviation is much higher than the previous dice noted above, and more interesting is the clear indication that the die is egg-shaped. This is visible by the axis on the 20 (and 1) and the 3 faces that surround it being positive numbers and the others negative.

Blue-large numbers

The deviation of axis lengths increases.

Blue-large numbers	mm	axis	diff from mean
20-1	18.47	1	-0.265
11-10	18.33	2	-0.405
13-7	18.81	3	0.075
14-6	18.84	4	0.105
16-4	18.66	5	-0.075
17-3	19.02	6	0.285
12-2	19.05	7	0.315
19-5	18.57	8	-0.165
15-9	18.8	9	0.065
18-8	18.8	10	0.065
SD	0.2291651515

Though not as obvious, there is some bulging here when you look at where the axes lie in relation to each other. What is interesting with this die is that despite being from the 1990s, the number distribution is not modern. For example, the opposite number from the 19 is a 5 (not a 2) and the opposite number from the 18 is an 8 (not a 3).

Pink

I don’t use it that often, but seeing these numbers I wonder if I ever will.

Pink	mm	axis	diff from mean
20-1	19.83	1	0.531
14-7	19.64	2	0.341
13-8	19.47	3	0.171
19-2	19.31	4	0.011
17-4	19.17	5	-0.129
15-6	19.25	6	-0.049
16-5	19.3	7	0.001
11-10	19	8	-0.299
12-9	18.96	9	-0.339
18-3	19.06	10	-0.239
SD	0.2804936125

Further deviation compared to the other die with elongation of the die along the first axis (like Blue-standard above) with modern number distribution.

Blue-bad paint

The worst offender by far.

Blue-bad paint	mm	axis	diff from mean
20-1	20.29	1	0.6609
14-7	19.99	2	0.3609
13-8	19.9	3	0.2709
19-2	19.68	4	0.0509
17-4	19.45	5	-0.1791
15-6	19.541	6	-0.0881
16-5	19.5	7	-0.1291
11-10	19.33	8	-0.2991
12-9	19.29	9	-0.3391
18-3	19.32	10	-0.3091
SD	0.3332521401

Like Pink and Blue-standard, this one has elongation of the die along the first axis. This is one of those dice I have come to use when, as a DM, I would rather not have a monster or trap hit a player’s character.

Conclusion

A perception of how dice perform without data to support that performance is a dangerous thing. I have always seen certain dice as consistently bad in that they never roll high or always seem to roll low, which in general translates to an overall skewed rolling result. The measurements I’ve taken of each of the 7 d20s seem to confirm what I perceived. However, despite the elongation of some of the dice (which Lou Zocchi mentions in the videos) there are 2 points which may make the shape data above not very meaningful.

1. The idea of an egg shaped die, having the numbers at the poles less likely to appear, isn’t a good analogy. An egg, a true egg, has a curved surface which causes it to try to achieve a rest state on its side. Dice however have faces which counteract forces that would cause egg-like rolling.
2. With the distances of the above deviations from the mean of the axis lengths being extremely small (being in the tenths or even hundredths of millimeters) and the mass of the die being quite small, such variations would likely not effect a die roll of the spherical-shaped d20 to a noticeable degree.

Despite these two points, the perceived performance of the dice and their axes deviations from the mean is certainly suggestive and cannot be completely ignored without complete rolling statistics for each die. Until then, I’m certainly going to stay away from those die that have an overall standard deviation above 0.084 mm.