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

3,2

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.

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

Table of n, a(n) for n=3..10.

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

Example

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.

Crossrefs

Cf. A110312, A110356.

Sequence in context: A306654 A368358 A114602 * A073922 A201945 A375319

Adjacent sequences: A109997 A109998 A109999 * A110001 A110002 A110003

Keyword

nonn

Author

N. J. A. Sloane, Sep 11 2005

Status

approved