History of Mathematics Project | Serpentinite Icosahedral Die

History of
Mathematics

Project

around 200 BCE–400 CE

Serpentinite Icosahedral Die

Ancient Egyptian 20-sided die

This 20-sided die with very precisely cut regular triangular faces dates to around 200 BCE–400 CE and is one of several known examples of ancient icosahedral dice bearing numerals on their faces.

Polyhedral dice made in various materials have survived from the Hellenistic and Roman periods from ancient Egypt. Most have faces labeled with the numbers 1 to 20 written either in Greek or Roman numerals. While the specific use of these dice is not known, it has been suggested they were used for games.

Artifact dimensions

3.2 cm × 3.8 cm × 3.4 cm

Artifact origin

Egypt

Current artifact location

Metropolitan Museum of Art, New York City

Catalog number

551072

Timeline

Interactive Content

Computational Explanation

As known to the ancient Greeks, there are exactly five convex solids composed of equal regular polygons placed in symmetrical positions. These are known as the Platonic solids:

In[]:=

GraphicsRow

Platonic solids

POLYHEDRA

["Graphics3D"]

Out[]=

One of the greatest achievements of Greek mathematics was its exact construction of the five Platonic solids using the classical tools of a straightedge and a compass.

The regular icosahedron is the Platonic solid having 20 equivalent equilateral triangular faces:

In[]:=

MeshCells[Icosahedron[],2]//Length

Out[]=

In[]:=

MeshCellCount[Icosahedron[],2]

Out[]=

A simple method of building an icosahedron out of paper or similar material consists of folding a net of equilateral triangles such as the following:

In[]:=

regular icosahedron

POLYHEDRON

["Net"]

Out[]=

The icosahedral die from the Metropolitan Museum of Art is strikingly regular and it is one of several known examples of ancient icosahedral dice with faces labeled with the Greek numerals ranging from 1 to 20:

In[]:=

Rule@@@Transpose[{StringDrop[IntegerString[Range[20],"Greek"],-1],Range[20]}]

Out[]=

{Α1,Β2,Γ3,Δ4,Ε5,Ϝ6,Ζ7,Η8,Θ9,Ι10,ΙΑ11,ΙΒ12,ΙΓ13,ΙΔ14,ΙΕ15,ΙϜ16,ΙΖ17,ΙΗ18,ΙΘ19,Κ20}

In order to construct a virtual copy, first extract the dominant colors from the image:

In[]:=

palette=DominantColors



Out[]=





To match the font, pick a similar sans serif font family:

In[]:=

Framed[greeks=Table[Style[StringDrop[IntegerString[n,"Greek"],-1],Last[palette],FontSize16,FontFamily"Thonburi"],{n,20}],BackgroundDarker@First[palette]]

Out[]=

Α,Β,Γ,Δ,Ε,Ϝ,Ζ,Η,Θ,Ι,ΙΑ,ΙΒ,ΙΓ,ΙΔ,ΙΕ,ΙϜ,ΙΖ,ΙΗ,ΙΘ,Κ

Now apply the Greek numerals to the faces of the die using Texture:

faces=Transpose(*Greeknumbersastextures*)TextureRasterizeFramed#,

styles

,"Image",

style

&/@greeks,(*PolygonscorrespondingtofaceswithappliedVertexTextureCoordinates*)FlattenNormal

regular icosahedron

POLYHEDRON

["Faces"]/.Polygon[p_]:>Polygon[N[p],VertexTextureCoordinates{{0,0},{1,0},{1/2,1.2}}];

… and view from various points above some of the vertices:

In[]:=

GraphicsGridPartitionTableGraphics3Dfaces,ViewPoint15i,

styles

,BaseStyleEdgeForm[Directive[Thick,palette〚3〛]],i,

regular icosahedron

POLYHEDRON

["VertexCoordinates"][[1;;All;;2]],3,

styles



Out[]=

Other Resources

History of Mathematics
Icosahedral Die with Divine Entities

History of Mathematics
Pottery Sherd Icosahedron Construction

Wolfram Demonstration
Euclid's Construction of a Regular Icosahedron (XIII .16)

Additional Reading

Horster, G. "Antike Polyeder: Vom Spiel mit Form und Zahl im Ptolemaïschen Ägypten zum Kleinod in Römischen Europa." In Jahrbuch des Deutschen Archäologischen Instituts, Vol. 132, p. 147, cat. III.B.14, 2017.
Senechal, M. (Ed.). Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination. Springer, p. 56, 2013.

Additional Links

Image Credits

Metropolitan Museum of Art