A110000 Minimal number of polygonal pieces in a dissection of a regular n-gon to an equilateral triangle (conjectured).

1, 4, 6, 5, 8, 7, 8, 7

Offset: 3

Author

N. J. A. Sloane, Sep 11 2005

Keywords

• nonn

Comments

I do not know which of these values have been proved to be minimal.

Turning over is allowed. The pieces must be bounded by simple curves to avoid difficulties with non-measurable sets.

Examples

			a(3) = 1 trivially.
a(4) <= 4 because there is a 4-piece dissection of an equilateral triangle into a square, due probably to H. Dudeney, 1902 (or possible C. W. McElroy - see Fredricksen, 1997, pp. 136-137). Surely it is known that this is minimal? See illustrations.
Coffin gives a nice description of this dissection. He notes that the points marked * are the midpoints of their respective edges and that ABC is an equilateral triangle. Suppose the square has side 1, so the triangle has side 2/3^(1/4). Locate B on the square by measuring 1/3^(1/4) from A, after which the rest is obvious.
For n >= 5 see the Theobald web site.
		

References

• G. N. Frederickson, Dissections: Plane and Fancy, Cambridge, 1997.
• H. Lindgren, Geometric Dissections, Van Nostrand, Princeton, 1964.
• H. Lindgren (revised by G. N. Frederickson), Recreational Problems in Geometric Dissections and How to Solve Them, Dover, NY, 1972.

Links

• Stewart T. Coffin, Dudeney's 1902 4-piece dissection of a triangle to a square, from The Puzzling World of Polyhedral Dissections.
• Stewart T. Coffin, The Puzzling World of Polyhedral Dissections, Chapter 1. (See section "Geometrical Dissections".)
• Geometry Junkyard, Dissection
• Gavin Theobald, Triangle dissections
• Vinay Vaishampayan, Dudeney's 1902 4-piece dissection of a triangle to a square

Crossrefs

Cf. A110312, A110356.