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Haga's TUP (Turned Up Part) origami of a square

Description

Move the red point to fold the bottom right corner of the square. What kinds of polygons are possible? Which are impossible? Why? What are their mathematical names and properties? You could focus either on the turned-up part of paper or the entire sheet of paper. Haga, K. (2008) Origamics: mathematical explorations through paper folding. Singapore: World Scientific. Translated by Fonacier, J. and Isoda, M. p.45-58