Topic
Equidissection
About: Equidissection is a research topic. Over the lifetime, 8 publications have been published within this topic receiving 45 citations.
Papers
TL;DR: This approach incorporates the techniques of Thomas, Monsky, and Mead, in particular, the use of Sperner's lemma and non-Archimedean valuations, but also makes use of affine transformations to distort a given regular polygon into one to which those techniques apply.
Abstract: This paper answers the question, "If a regular polygon withn sides is dissected intom triangles of equal areas, mustm be a multiple ofn?" Forn=3 the answer is "no," since a triangle can be cut into any positive integral number of triangles of equal areas. Forn=4 the answer is again "no," since a square can be cut into two triangles of equal areas. However, Monsky showed that a square cannot be dissected into an odd number of triangles of equal areas.
We show that ifn is at least 5, then the answer is "yes." Our approach incorporates the techniques of Thomas, Monsky, and Mead, in particular, the use of Sperner's lemma and non-Archimedean valuations, but also makes use of affine transformations to distort a given regular polygon into one to which those techniques apply.
19 citations
TL;DR: In this paper, it was shown that a centrally symmetric regular polygon cannot be cut into an odd number of triangles of equal areas, even if the restriction "regular" is removed from the hypothesis.
Abstract: In 1970 Monsky proved that a square cannot be cut into an odd number of triangles of equal areas. In 1988 Kasimatis proved that if a regularn-gon,n ⩾ 5, is cut intom triangles of equal areas, thenm is a multiple ofn. These two results imply that a centrally symmetric regular polygon cannot be cut into an odd number of triangles of equal areas. We conjecture that the conclusion holds even if the restriction “regular” is deleted from the hypothesis and prove that it does forn = 6 andn = 8.
10 citations
TL;DR: The first result of this type was obtained by Paul Monsky in 1970 as discussed by the authors, who proved that a square cannot be cut into an odd number of triangles of equal areas, and in 2000, Sherman Stein conjectured that the same holds for any balanced polygon.
Abstract: In this paper, we show that a lattice balanced polygon of odd area be cut into an odd number of triangles of equal areas. The first result of this type was obtained by Paul Monsky in 1970. He proved that a square cannot be cut into an odd number of triangles of equal areas. In 2000, Sherman Stein conjectured that the same holds for any balanced polygon. We also show between the equidissection problem and tropical geometry. Bibliography: 9 titles.
4 citations
TL;DR: This work investigates equidissections of a trapezoid T(a), where the ratio of the lengths of two parallel sides is a, and shows that if n is large enough, n is in S(T(a)) iff n/(1+a) is an algebraic integer.
4 citations
TL;DR: In this paper, it was shown that a triangle cannot be cut into an odd number of triangles of equal areas, and it was proved that the same is true for any centrally symmetric polygon.
Abstract: In 1970 Monsky proved that a square cannot be cut into an odd number of triangles of equal areas. In 1990 it was proved that the statement is true for any centrally symmetric polygon. In the present paper we consider dissections of general polygons into triangles of equal areas.
3 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2020 | 1 |
| 2014 | 1 |
| 2013 | 1 |
| 2012 | 1 |
| 2008 | 1 |
| 2003 | 1 |