The 759 octads of the Steiner system S(5,8,24) are displayed

rather neatly in the Miracle Octad Generator of R. T. Curtis.

A March 9, 2018, construction by Iain Aitchison* pictures the

759 octads on the faces of a *cube* , with octad elements the

24 edges of a *cuboctahedron *:

The Curtis octads are related to symmetries of the *square*.

See my webpage "Geometry of the 4×4 square" from March 2004.

Aitchison's p. 42 slide includes an illustration from that page —

*Aitchison's* octads are instead related to symmetries of the *cube*.

Note that essentially the same model as Aitchison's can be pictured

by using, instead of the 24 edges of a cuboctahedron, the 24 *outer*

faces of subcubes in the *eightfold cube* .

** Image from Christmas Day 2005.**

* http://www.math.sci.hiroshima-u.ac.jp/branched/files/2018/

presentations/Aitchison-Hiroshima-2-2018.pdf.

See also Aitchison in this journal.