This design can be visualized by a tetrahedron:

The steps of construction are the following:

The "recipe" for the group selection "solids" in DISCRETA is the following:

- take a tetrahedron
- take the edge midpoints of the tetrahedron (by pressing "Tetrahedron", "choose", "midpoints of edges" and again "choose") - this gives an inscribed octahedron
- take the dual (which is also a tetrahedron, whose vertices are the midpoints of the faces of the original tetrahedron) by pressing "Tetrahedron", "choose", "dual" and "choose"
- last thing to do is to choose "add central point"

The results of each single step is coloured in pink at the pictures above.

This construction yields a group derived from the symmetry group A_4 (group order: 12) of the tetrahedron (the group is induced on the edge midpoints and on the dual tetrahedron, and finally you can add a fixpoint such that you get a group on 15 points). Under the action of this group you receive by computations with DISCRETA 6 2-(15, 7, 3) designs (some of them are isomorphic under a larger group). More information on them you can read in this report.

One of the designs is visualized below: you get the following orbit representants coloured
in pink (with the corresponding orbit length). Adding the orbit lengths, we get
15 blocks as recommended.

orbit length: 4 | orbit length: 6 | orbit length: 4 | orbit length: 1 |

When we consider the 2-subsets and the 3-subsets of the 15 points above and fix lambda=1, we receive the following Kramer-Mesner matrix w.r.t. the action of the same group(size: 13x46):

0 3 1 0 0 3 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 2 2 0 2 2 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 2 1 2 2 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 1 2 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 1 0 1 1 0 0 0 0 2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 0 3 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 1 1 1 0 0 1 1 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 2 0 2 0 0 0 0 4 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 2 1 0 1 1 1 1 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 1 0 1 2 1 0 1 1 1 0

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 1 2 2 1 2

It is very interesting that we can visualize the projective plane PG(3,2) with this construction of the tetrahedron - it is a Steiner triple system on 15 points. By solving the corresponding matrix equation, one gets 12 designs with DISCRETA, the orbits of one of them are the following (see also the report):

orbit length: 6 | orbit length: 12 | orbit length: 4 |

orbit length: 4 | orbit length: 6 | orbit length: 3 |

Therefore we get 35 blocks. It is a very interesting remark that all the blocks of the point-hyperplane design above consist of Fano planes whose blocks are exactly blocks from the PG(3,2)!

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Last updated: June 16, 1999.