The Fano tetrahedron

The "Fano tetrahedron" is built from Fano planes. We start with a description of the Fano plane, which you can skip if you're already familiar with it.

The Fano plane

 Diagram 1. The Fano plane

A Fano plane is is a finite projective plane of seven points and seven lines, with three points on every line and three lines through every point. It is a balanced block design, with seven blocks (the lines) and seven elements (the points). It can be specified as the balanced block design 2-(7,7,3,3,1); and as the last value of this balanced block design is 1, it is a Steiner system, and as such its descriptor can be written S(2,3,7). Any two points lie on one and only one line, and any two lines pass through one and only one point.

A Fano plane is shown to the right. Three of the lines are shown in dark gray, the in red, and one in green – this has no mathematical significance, but it may make the diagram easier to understand. Its vertices are labelled with numbers, which should be regarded, not as integers, but as bit-strings (001, 010, 011 etc.). Wherever two points on a line are labelled p and q, the third point on that line is labelled p xor y.

Building the Fano tetrahedron

 Diagram 2a. A tetrahedral net built from four Fano planes Diagram 2b. The tetrahedral net with six more lines added.

Just as a tetrahedron is built from four triangles, a Fano tetrahedron is built from four Fano planes. First we build the "net" of the tetrahedron, as shown in diagram 2a. A tetrahedron can be built from this net by folding it along the lines labelled 3, 5 and 6, and uniting the pairs and triplet of vertices with the same label.

But what we have now is not a balanced block design or Steiner system, for two reasons. We need to add one more point, labelled 15, in the centre of the tetrahedron, with seven lines through it, joining each vertex (1,2,4,8) to its antipodal face (14,13,11,7) and each edge (3,5,6) to its antipodal edge (12,10,9). Also we need to add six more lines, shown in diagram 2b in blue and purple.

The diagrams do not show the central vertex with its label "15". You'll have to imagine it, and the lines that pass through it.

As with the Fano plane in diagram 1, each pair of points lies on one line. It is therefore a Steiner syetem, S(2,3,15). The labels on the points are again such that the xor of any two points is the label on the third point on their line.

Catalogue of lines

Below is a table cataloging the 35 lines (blocks) of the Fano tetrahedron, and listing the elements (Vertices, Edges, Faces, and the Centre) of the tetrahedron that lie on them. The others are easy to understand, but the blue and magenta lines may seem irregular. In fact they join a pair of faces and the edge antipodal to their mutual edge. The blue edges and the magenta ones are equivalent, but look different when drawn on the net in the diagram.

Colour of lines
in diagram
Elements
on line
Number of
such lines
Description
GreyV E V6edges
RedV F E12medians
GreenE E E4in-circles
BlueF E F6lie on great circles
of the tetrahedron
Magenta
(not shown)V C F4pass through the centre,
joining antipodes
E C E3
Total35

Another view

 Diagram 3. No longer a net, but the tetrahedron drawn with one vertex (its apex, if you like) at infinity in all directions.

Diagram 3 shows another view of the Fano tetrahedron. Each edge is now shown only once. The apex vertex, labeled "8", is at infinity.

The magenta lines, which lie along great circles, ought to be straight, but would then coincide with red lines. So instead, they have been drawn with slight S-shaped curves.

Further simplexes

Just as we can use the Fano plane to build the Fano tetrahedron, we can use the Fano tetrahedron to build the Fano 5-cell, with 5 vertices, 31 numbered points, and 155 lines, the Steiner system S(2,3,31); and so on.