Abstract
Let A and B be convex polygons. We say that A and B are D-equivalent if there are convex polygons A = A 1,A 2,...,A n = B and Dudeney dissections of A i to A i + 1 (1 ≤ i ≤ n − 1). A polygon is called a tile if the 2-dimensional Euclidean plane can be tiled by congruent copies of the polygon. A polygon is called a normal tile if the plane can be tiled by congruent copies of the polygon which are obtained without turning over the polygon. The numbers of types of convex tiles and convex normal tiles are still uncertain. In this paper, we prove that all convex normal tiles with the same area that we know so far are D-equivalent.
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References
Akiyama, J., Nakamura, G.: Congruent Dudeney dissections of triangles and convex quadrilaterals - All hinge points interior to the sides of the polygons. Algorithms and Combinatorics 25, 43–64 (2003)
Akiyama, J., Nakamura, G.: Determination of all convex polygons which are chameleons. IEICE TRANS. Fundamentals E86-A(5), 978–986 (2003)
Dudeney, H.E.: The Canterbury Puzzles. Dover, New York (2002); originally published by Heinemann, W., London (1907)
Frederickson, G.N.: Hinged Dissections: Swinging & Twisting. Cambridge University Press, Cambridge (2002)
Grunbaum, B., Shephard, G.C.: Tilings and Patterns. Freeman, New York (1986)
Grunbaum, B., Shephard, G.C.: Some Problems on Plane Tilings. In: Mathematical Recreations: A Collection in Honor of Martin Gardner, pp. 167–196. Dover, New York (1998); Originally published: Klarner, D.A. (ed.) The Mathematical Gardner, Wadsworth International, Belmont, California, (1981)
Kershner, R.B.: On paving the plane. American Mathematical Monthly 75, 839–844 (1968)
Niven, I.: Convex polygons that cannot tile the plane. American Mathematical Monthly 85, 785–792 (1978)
Stein, R.: A New Pentagon Tiler. Mathematics Magazine 58(5), 308 (1985)
Schattschneider, D.: Tiling the Plane with Congruent Pentagons. Mathematics Magazine 51, 29–44 (1978)
Schattschneider, D.: In Praise of Amateurs. In: Mathematical Recreations: A Collection in Honor of Martin Gardner, pp. 140–166. Dover, New York (1998); Originally published: Klarner, D.A. (ed.)The Mathematical Gardner, Wadsworth International, Belmont, California (1981)
Taylor, H.M.: On some geometrical dissections. Messenger of Mathematics 35, 81–101 (1905)
The 14 Different Types of Convex Pentagons that Tile the Plane, http://www.mathpuzzle.com/tilepent.html
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Akiyama, J., Kobayashi, M., Nakamura, G. (2008). Dudeney Transformation of Normal Tiles. In: Ito, H., Kano, M., Katoh, N., Uno, Y. (eds) Computational Geometry and Graph Theory. KyotoCGGT 2007. Lecture Notes in Computer Science, vol 4535. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89550-3_1
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DOI: https://doi.org/10.1007/978-3-540-89550-3_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-89549-7
Online ISBN: 978-3-540-89550-3
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