Origami

## Foldable maps (in construction)

• studying why and how flat maps distort features of a spherical surface
• origami, Japanese word for the now international art of folding recognizable or abstract figures in plain paper
• creating a pseudoglobe in the shape of a polyhedron

#### Fujimoto's Cube

Perhaps the most elegant origami cube, the model created by the great folder and researcher Shuji Fujimoto requires a single square sheet of paper. You can print and fold a sheet creating a nice cubical world "globe". Cut the big square and follow the small crease pattern included (it is inside the square but will not appear in the final model) showing valley and mountain folds.
Experienced folders should have no difficulty folding the cube using the crease pattern alone, but detailed diagrams can be found in at least two books (I doubt I can draw better ones), [KT87] and [Ja90], pages 38-39. Thoki Yenn includes diagrams at his site, in the context of the famous Delian problem. As soon as I get the time, I plan drawing a map pattern for Kazuo Haga's octahedron, also diagrammed in Origami for the Connoisseur.

#### Montroll's Cube

 Preprinted pattern for Montroll's cube
More recently, John Montroll devised several origami polyhedra folded from a single sheet. His cube ([Mon02], pages 14-15) is particularly elegant: all faces comprise a single uncreased piece. Warning: this is a larger file (560 KB); before printing, make sure it fits your paper.

#### Haga-Kasahara Cube

A cube superficially similar to Montroll's was previously invented independently by Haga and Kunihiko Kasahara. It can be folded from the same printed pattern for Montroll's, using diagrams in [KT87], pages 58-59, mirrored left-to-right.
 Crease patterns for Haga-Kasahara (left) and Montroll (right) cubes: pale blue marks the external faces. Edge "A" should be on the top at step 4 of [KT87] (all steps should be mirrored left-to right); edge "B" should be on the top at step 5 of [Mon02].

### Miura-ori

Another interesting marriage of paperfolding and cartography (or at least practical map usage) is presented by the Miura-ori technique (named after its pioneer Koryo Miura): folding a large sheet in a way which allows instant opening by just pulling apart two corners. Collapsing the map back is equally fast and easy. A similar method ensures deployment of solar panels in spacecraft.

Almost identical diagrams for a Miura-ori sample are included in [ORU5], pages 34-37 and [Kan97], pages 236-238.

 www.progonos.com/furuti    June 16, 2018