Dividing a square into similar rectangles (or, equivalently, tiling a square with similar rectangles) is a problem in mathematics.
Dividing a square into similar rectangles (or, equivalently, tiling a square with similar rectangles) is a problem in mathematics.
There is only one way (up to rotation and reflection) to divide a square into two similar rectangles.
However, there are three distinct ways of partitioning a square into three similar rectangles: [1] [2]
The fact that a rectangle of aspect ratio ρ2 can be used for dissections of a square into similar rectangles is equivalent to an algebraic property of the number ρ2 related to the Routh–Hurwitz theorem: all of its conjugates have positive real part. [3] [4]
In 2022, the mathematician John Baez brought the problem of generalizing this problem to n rectangles to the attention of the Mathstodon online mathematics community. [5] [6]
The problem has two parts: what aspect ratios are possible, and how many different solutions are there for a given n. [7] Frieling and Rinne had previously published a result in 1994 that states that the aspect ratio of rectangles in these dissections must be an algebraic number and that each of its conjugates must have a positive real part. [3] However, their proof was not a constructive proof.
Numerous participants have attacked the problem of finding individual dissections using exhaustive computer search of possible solutions. One approach is to exhaustively enumerate possible coarse-grained placements of rectangles, then convert these to candidate topologies of connected rectangles. Given the topology of a potential solution, the determination of the rectangle's aspect ratio can then trivially be expressed as a set of simultaneous equations, thus either determining the solution exactly, or eliminating it from possibility. [8]
As of March 2023 [update] , the following results (sequence A359146 in the OEIS ) have been obtained for the number of distinct valid dissections for different values of n: [7] [9] [10]
| n | # of dissections |
|---|---|
| 1 | 1 |
| 2 | 1 |
| 3 | 3 |
| 4 | 11 |
| 5 | 51 |
| 6 | 245 |
| 7 | 1372 |
| 8 | 8522 |
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Circle packing in a square is a packing problem in recreational mathematics, where the aim is to pack n unit circles into the smallest possible square. Equivalently, the problem is to arrange n points in a unit square aiming to get the greatest minimal separation, dn, between points. To convert between these two formulations of the problem, the square side for unit circles will be L = 2 + 2/dn.
A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are based on it, explaining its name. It is commonly used as a pattern for floor tiles. When used for this, it is also known as a hopscotch pattern or pinwheel pattern, but it should not be confused with the mathematical pinwheel tiling, an unrelated pattern.
In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by Martin Gardner in his "Mathematical Games" column in the May 1963 issue of Scientific American. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in Mathematics Magazine.
In geometry, an equidissection is a partition of a polygon into triangles of equal area. The study of equidissections began in the late 1960s with Monsky's theorem, which states that a square cannot be equidissected into an odd number of triangles. In fact, most polygons cannot be equidissected at all.
David Anthony Klarner was an American mathematician, author, and educator. He is known for his work in combinatorial enumeration, polyominoes, and box-packing.
Edmund Orme Harriss is a British mathematician, writer and artist. Since 2010 he has been at the Fulbright College of Arts & Sciences at The University of Arkansas in Fayetteville, Arkansas where he is an Assistant Professor of Arts & Sciences (ARSC) and Mathematical Sciences (MASC). He does research in the Geometry of Tilings and Patterns, a branch of Convex and Discrete Geometry. He is the discoverer of the spiral that bears his name.
