# MR88k: 52020 Pérez-Gómez, R

## The four regular mosaics missing in the Alhambra

Alhambra, *Computers and Mathematics with Applications*
**14** (1987), 133-137; see also
After careful observation of the Alhambra in Granada, B
Grünbaum, Z Grübaum and G C Shepherd [same journal, Part
**B** **12** (1986), no. 3-4, 64-643 = I Hargittai, (ed.),
*Symmetry Unifying Human Understanding* Pergamon 1986 (1986);
MR87k: 05056] concluded that, although the Moorish decorators used 13 of
the 17 wallpaper groups, they never used the remaining four, namely
`p2`, `pgg`, `pg`, `p3m1`. The author
claims
to have found evidence of the missing groups. His figure 1 does seem to
establish half-turens that generate `p2`, but Figure 2, which is
intended to be helpful, is incorrectly drawn. (Compare the crossed
strips at *D* with the corresponding pairs higher up.) Figure 4
clearly has the group `pgg`, and Figure 6 has `pg`, But
his claim for `p3m1` (which is generated by reflection in three
sides of an equilateral triangle, not a rotation and two translations)
is questionable. It may now be reasonably asserted that, among the 17
groups, the Alhambra lacks only `p3m1`. On the other hand, B
Grünbaum *et al.* [*op. cit.*, Figure 8] found this
group in a wall decoration from a synagogue in Toledo (Spain).

H S M Coxeter (Toronto, Ont.)

We can simplify the essence of `p2` to (note the bar in the
middle):

This page is maintained by
Peter M Lee

*Revised 14 June 2005*