The XVII Annual Colloquium of the International Society for Board Game Studies April 2015

Astragali: some observations on the anatomy of different mammalian species and their performance as 4-sided dice when thrown onto different surfaces in the form of either the original bone or 3-D printed models of them

1Duggan E, 1Mortimore D, 2Emmens D, 1Taylor D and 1Bell GD 1University Campus Suffolk, Ipswich, and 2Department of Medical

Physics Ipswich Hospital Medical Trust, Ipswich IP5 4PD

Astragali: some observations on the anatomy of different mammalian species

Fossilized deer Astragali, CT scanned to produce STL files for 3-D printing.

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Astragali: some observations on the anatomy of different mammalian species

X-ray CT scan of two bone sheep astragali and one deer astragalus made on a 3D printer from PLA. On the left is a Water phantom.

Sheep astragali

Astragali: some observations on the anatomy of different mammalian species

Human Talus segmented out from CT scan of a patient

Astragali: some observations on the anatomy of different mammalian species

• Only ones of interest to this Society are those of use as 4-sided dice or divination. • All the astragali used in the Ancient and Classical World as

randomisation agents come from the hind legs (hocks) of even-toed artiodactyls ungulates such as sheep, goats or deer but not pigs.

• We will ignore larger astragali from animals such as bovines

which are basically binary.

• Uneven toed ungulates such as horses were not used.

• Plantigrade animals such as primates and Digigrade animals such as dogs and cats all have unsuitable shaped (or sized) bones to make satisfactory

4-sided dice.

Astraguli can only land with one of 4 different surfaces facing upwards

Sinuous Flat Convex Concave

Astraguli can only land with one of 4 different surfaces facing upwards

Sinuous Flat Convex Concave

The Convex and Concave surfaces being wider and broader come up significantly more frequently than the narrower Sinuous or flat surfaces.

The Convex and Concave surfaces being wider and broader come up significantly more frequently than the narrower Sinuous or flat surfaces. But what does the literature say?

From a tossing of a modern sheep’s astragalus the empirical probabilities were approximately one in ten each of throwing a 1 or of throwing a 6 and about four in ten, of throwing a 3 or a 4… Best reference we could find was that of David F.N. Games, Gods & Gambling 1962. Page 7. Publishers Bell and Bian Ltd, Glasgow

David states, for every 10 throws :- Convex 4/10 Concave 4/10 Sinuous 1/10 Flat 1/10

Mentions possible effect of throwing the dice onto different surfaces but gives no indication as to the number of throws/observations, she bases her 4/10, 4/10, 1/10 and 1/10 figures

Purpose of our study

To find out: • The relative frequency the 4 astragali surfaces appear

uppermost when using sheep bones (left vs right and larger vs smaller) on either a carpeted surface or a hard smooth wooden surface.

• How good were astragali made out of PLA on a 3D printer when compared with authentic bone ones?

Table 1. Two smaller sheep astragali A (left) and B (right) thrown onto a carpeted surface

Side A (n=500) B (n=500) Total of A+B (n=1000)

Per 100 throws Odds

1 (concave) Volcanus 198 210 408 40.8 .408

2 (convex) Aquila 199 196 395 39.5 .395

3 (sinuous) Caesar 51 50 101 10.1 .101

4 (flat) Canis 52 44 96 9.6 .096

500 500 1000

Table 2. 3D printed PLA model of a) deer astragalus and b) sheep astragalus thrown onto a carpeted surface

Side 3D printed PLA deer astragalus (n=500)

3D printed PLA sheep astragalus (n=500)

Total of 3D PLA printed deer and sheep astragali (n=1000)

Per 100 throws Odds

1 (concave) Volcanus 194 206 400 40.0 .400

2 (convex) Aquila 206 180 386 38.6 .386

3 (sinuous) Caesar 46 62 108 10.8 .108

4 (flat) Canis 54 52 106 10.6 .106

500 500 1000

Table 3. Combination of two smaller sheep astragali and the 3D printed models of both deer and sheep astragali thrown onto a carpeted surface

Side Sheep bone astragali Total of A+B (n=1000)

Total of 3D printed PLA deer and sheep astragalus (n=1000)

Total of sheep bone and 3D printed PLA model astragali (n=2000)

Per 100 throws Odds

1 (concave) Volcanus 408 400 809 40.4 .404

2 (convex) Aquila 395 386 781 39.04 .390

3 (sinuous) Caesar 101 108 209 10.5 .105

4 (flat) Canis 96 106 202 10.1 .101

1000 1000 2000

Table 4. Two smaller sheep astragali A and B thrown onto an oak table surface

Side A (n=1000) B (n=1000) Total of A+B (n=2000)

Per 100 throws Odds

1 (concave) Volcanus 378 406 784 39.2 .392

2 (convex) Aquila 445 474 919 45.9 .459

3 (sinuous) Caesar 80 54 134 6.7 .067

4 (flat) Canis 97 66 163 8.15 .082

1000 1000 2000

Table 5. Two larger sheep astragali (left and right) thrown onto an oak table surface

Side A Left (n=500) B Right (n=500) Total of left + right (n=1000)

Per 100 throws Odds

1 (concave) Volcanus 199 193 392 39.2 .392

2 (convex) Aquila 222 224 446 446 .446

3 (sinuous) Caesar 29 44 73 73 .073

4 (flat) Canis 50 39 89 89 .089

500 500 1000

Table 6. Combined results of the two larger sheep astragali (left and right) and two smaller sheep astragali (A+B) thrown onto an oak table surface

Side Two larger sheep astragali - left + right (n=1000)

Two smaller sheep astragali - A + B (n=2000)

Total of all 4 sheep astragali (n=3000)

Per 100 throws Odds

1 (concave) Volcanus 392 784 1176 39.2 .392

2 (convex) Aquila 446 919 1365 45.5 .455

3 (sinuous) Caesar 73 134 207 6.9 .069

4 (flat) Canis 89 163 252 8.4 .084

1000 2000 3000

Table 7. Combined results of all sheep bone astragali thrown onto an oak table surface compared with all sheep bone and 3D printed model PLA astragali thrown onto a carpet surface

Side Oak table total of all 4 sheep astragali (n=3000)

Chances per 100 throws

Carpet surface, total of sheep bone and 3D printed PLA model astragali (n=2000)

Per 100 throws

1 (concave) Volcanus 1176 39.2 808 40.4

2 (convex) Aquila 1365 45.5 781 39.0

3 (sinuous) Caesar 207 6.9 209 10.5

4 (flat) Canis 252 8.4 202 10.1

3000 2000

Table 7. Combined results of all sheep bone astragali thrown onto an oak table surface compared with all sheep bone and 3D printed model PLA astragali thrown onto a carpet surface

Side Oak table total of all 4 sheep astragali (n=3000)

Chances per 100 throws

Carpet surface, total of sheep bone and 3D printed PLA model astragali (n=2000)

Per 100 throws

1 (concave) Volcanus 1176 39.2 NS 808 40.4

2 (convex) Aquila 1365 45.5 p<0.0001 781 39.0

3 (sinuous) Caesar 207 6.9 p<0.0001 209 10.5

4 (flat) Canis 252 8.4 p<0.0394 202 10.1

3000 2000

Conclusions

• The hocks of even-toed ungulates such as sheep, goats and deer were used to provide the astragalus or talus used for

gaming and divination.

• Life-sized Astragali made from 3D printed PLA behave very similarly to those made of the bone of either a deer or sheep.

• Astragali thrown onto a carpeted surface show significant differences when compared with the same dice thrown onto a smooth hard wooden surfaces in terms of the frequency that each of the 4 sides appears uppermost.

Extra slide

Statistics on Table 7 Chi-squared 2 X 4 gives a Chi squared of 34.12 and with 3 D of F, p <0.001. Chi-squared 2 X 2 for side 1, side 2, side 3 and side 4 gave Chi-squared (with Yates correction) values of 0.731, 19.800, 19.360 and 4.234 respectively. The corresponding p values for sides 1-4 being p=.3923(NS), p<0.0001, p< 0.001 and p < 0.0394 respectively