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.
Read more about this topic: Orbifold Notation
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