Icosahedron's Primary Great Circle Data

The icosahedron has 3 goups of spin axes. These are

  1. 6 vertex-to-vertex axes
  2. 15 midedge-to-mid-edge axes
  3. 10 face-center-to-face-center axes
for a total of 31 great cricles. Each group of axes defines a different set of great circles which criss-cross the surface of the sphere and therefore the faces of the spherical icosahedron as well. Below is an illustration showing a single spherical icosahedron face criss-crossed with the great circle arcs. The spherical triangle in bold is the LCD spherical triangle. (More on this below.)

Below I show which axis group grenerates which great circle arcs across the icosahedron's faces. The bold arcs are generated by the associated axis group as labeled.

The surface of the spherical icosahedron can be covered by 60 "positive" and 60 "negative" (total = 120) spherical triangles known as the icosahedron LCD spherical triangle. This is the spherical triangle shown in bold in the above illustration.

The next illustration gives the icosahedron's LCD triangles labels for the following tables.

Now for the tables of the spherical icosahedron's LCD spherical triangle angles.

Icosahedron's Surface LCD Angular Data
CAD60.000 000 000asin(3,3)
EBF36.000 000 000
ACD31.717 474 411
DCE37.377 368 141
ECF20.905 157 44845+acos(1,2)
ADC90.000 000 000acos(0,1)
CDE90.000 000 000acos(0,1)
BEF54.735 610 317acos(2,3)
CED54.735 610 317acos(2,3)
CEF70.528 779 366asin(8,8)
BFE90.000 000 000acos(0,1)
CFE90.000 000 000acos(0,1)

Icosahedron's LCD Central Angular Data
AC20.905 157 44845+acos(1,2)
AD10.812 316 964
BE13.282 525 589
BF10.812 316 964
CD18.000 000 000
CE22.238 756 093acos(15,24)-30
CF20.905 157 44845+acos(1,2)
DE13.282 525 589
EF 7.761 243 907

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