Orbifold Notation - The Euler Characteristic and The Order

The Euler Characteristic and The Order

The Euler characteristic of an orbifold can be read from its Conway symbol, as follows. Each feature has a value:

• n without or before an asterisk counts as

• n after an asterisk counts as

• asterisk and count as 1

• counts as 2.

Subtracting the sum of these values from 2 gives the Euler characteristic.

If the sum of the feature values is 2, the order is infinite, i.e., the notation represents a wallpaper group or a frieze group. Indeed, Conway's "Magic Theorem" indicates that the 17 wallpaper groups are exactly those with the sum of the feature values equal to 2. Otherwise, the order is 2 divided by the Euler characteristic.