> Regarding puzzle 99 and 100:

The rotational and reflectional symmetries of a geometric object are given by the [Dihedral group](https://en.wikipedia.org/wiki/Dihedral_group) on that object.

I want to get more specific.

In the case of **Puzzle 99**, since we only have one object we know the category is a monoid as per **Puzzle 98**. Since every reflection or rotation has an inverse, this monoid must be a group.

Specifically, the group here is the [Dihedral 8 Group](https://en.wikipedia.org/wiki/Dihedral_group#Matrix_representation) \\(D_8\\) (using algebraic indexing). This group has 8 elements, so to answer part of **Puzzle 99**: *there are 8 morphisms in that category*.

In the case of **Puzzle 100**, I think the objects *represent* the morphisms in **Puzzle 99**. So they aren't quite the same.

Perhaps **Puzzle 100** is a vehicle for looking a topic in Category Theory that's missing in *Seven Sketches*?

The rotational and reflectional symmetries of a geometric object are given by the [Dihedral group](https://en.wikipedia.org/wiki/Dihedral_group) on that object.

I want to get more specific.

In the case of **Puzzle 99**, since we only have one object we know the category is a monoid as per **Puzzle 98**. Since every reflection or rotation has an inverse, this monoid must be a group.

Specifically, the group here is the [Dihedral 8 Group](https://en.wikipedia.org/wiki/Dihedral_group#Matrix_representation) \\(D_8\\) (using algebraic indexing). This group has 8 elements, so to answer part of **Puzzle 99**: *there are 8 morphisms in that category*.

In the case of **Puzzle 100**, I think the objects *represent* the morphisms in **Puzzle 99**. So they aren't quite the same.

Perhaps **Puzzle 100** is a vehicle for looking a topic in Category Theory that's missing in *Seven Sketches*?