Squaring the Square

Today I turned 26234 days old and one of the properties of the number 26234 is that it is a member of OEIS A217156 with \(n\)=30.

A217156

Number of perfect squared squares of order \(n\) up to symmetries of the square.

The comments in the OEIS entry go on to say that:

a(\(n\)) is the number of solutions to the classic problem of 'squaring the square' by \(n\) unequal squares. A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. The order of a squared rectangle is the number of constituent squares. A squared rectangle is simple if it does not contain a smaller squared rectangle, and compound if it does.

For \(n\)<21, it's not possible to put unequal squares together to form a larger square but for \(n\)=21, one such square is possible (see Figure 1):

Figure 1: source
The tiny square in the middle has a side of 2 units

This site puts this square (with a side of 112) in a historical context:

"The dissection was found in the night of March 22, 1978 with the aid of the DEC-10 computer of the Technological University Twente, The Netherlands. Since no simple perfect squared squares were found of orders less than 21, it is a simple perfect squared square of lowest order . Also, it is the only simple perfect squared square of order 21."

The acronym SPSS is often used in referring to these sorts of squares and the letters stand for Simple Perfect Squared Squares. The entries in OEIS A217156 are for both simple and compound squared squares. It is in fact the sequence represents the sum of two separate sequences so that a(\(n\)) = A006983(\(n\)) + A217155(\(n\)) representing numbers of simple and compound squares respectively. The progression from 21 up to 30 is shown in Figure 2:

Figure 2: A006983 + A217155 = A217156

The site squaring.net has PDF file for all the configurations of orders up to 236. The eight configurations for \(n\)=22 have sides of 110, 110, 139, 147, 154, 172, 172 and 192. The SPSS for the side of 192 is shown in Figure 3.

Figure 3: source

Here is a Numberphile video that delves into the history of these squares:

There is an auxiliary video to this which shows the first square discovered and another where the differences in sizes between smallest and largest are least extreme.

Perfect rectangles consist of squares of different sizes that fit together to form a rectangle. No square can be broken down further. The minimum required is 9 and there are two possible configurations. See Figures 4 and 5.

Figure 4: source
33 x 32 rectangle composed of 9 squares
where the tiny square has a side of 1 unit

Figure 5: source
69 x 61 rectangle composed of 9 squares
where the tiny square has a side of 2 units

The Plastic Number

One of the delights of investigating the number of one's diurnal age, as I'm addicted to doing, is that every so often something fascinating pops up. Today, day number 25996, was one such day. The Online Encyclopedia of Integer Sequences (OEIS) didn't throw up much of interest. However, my newly discovered Sequence Database did reveal an interesting looking sequence, although one with an unprepossessing name: Sequence vbihjo41hob4e. It runs thus, up to 25996:

0, 3, 5, 2, 7, 6, 8, 12, 13, 19, 24, 31, 42, 54, 72, 95, 125, 166, 219, 290, 384, 508, 673, 891, 1180, 1563, 2070, 2742, 3632, 4811, 6373, 8442, 11183, 14814, 19624, 25996, ...

It arises from the recurrence relationship:$$a_n=a_{n-2}+a_{n-3}-1 \text{ where } a_0=0, a_1=3, a_2=5 \text { and } n \geq 3$$Of course, it's reminiscent of the Fibonacci sequence in that it adds two previous terms to get the next one, although it skips the immediately preceding term and also subtracts 1. I knew that the ratio of one term to the preceding term in the Fibonacci sequence approaches the Golden Ratio and so I investigated the ratio for this sequence. I found that:$$a_{101}:a_{100} \approx 1.32471795724460$$This intrigued me and I wondered if this number could be an approximation to some mathematical constant. Fortunately, I have a link to a website called Mathematical Constants and Sequences and it didn't take me long to discover what this constant was. See Figure 1 (click to enlarge).

Figure 1

As can be seen, 1.324717957244746025960908 ... is referred to as the plastic number \( \rho \) or silver constant. It's the solution to the polynomial equation \(x^3=x+1\) which has the following real solution:$$\sqrt[\leftroot{-2} \uproot{10} 3]{\frac{9 + \sqrt{69}}{18}} + \sqrt[\leftroot{-2} \uproot{10} 3]{\frac{9 - \sqrt{69}}{18}}$$The essential recurrence relationship is that of:$$a_n=a_{n-2}+a_{n-3} \text{ with } n \geq 3$$If we set \(a_0=a_1=a_2=1\), then we have the Padovan sequence: 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, ... which is OEIS A000931.

My Sequence vbihjo41hob4e is essentially the Padovan sequence with different starting values \(a_0=0, a_1=3, a_2=5\) versus \(a_0=a_1=a_2=1\) and with 1 subtracted. The subtraction just means that it takes the numbers longer to get larger but addends or subtrahends and different starting values don't alter the long term behaviour. The ratio of one term to its predecessor always approaches the plastic number or silver constant. Subtracting integers greater than 1 leads to negative terms but the ratio still holds. Here is a permalink to SageMathCell that allows you experiment with the effect of different values.

The plastic number occurs in a variety of situations. Figure 2 shows one of them.

Figure 2: Three partitions of a square into similar rectangles

There are precisely three ways of partitioning a square into three similar rectangles:

The trivial solution given by three congruent rectangles with aspect ratio 3:1. The solution in which two of the three rectangles are congruent with the third one of twice the linear dimension of the congruent pair and where the rectangles have aspect ratio 3:2. The solution in which the three rectangles are mutually non congruent (all of different sizes) and where they have aspect ratio \( \rho^2 \). The ratios of the linear sizes of the three rectangles are: \( \rho \) (large : medium); \( \rho^2 \)(medium : small); and \( \rho^3 \) (large : small). The internal, long edge of the largest rectangle (the square's fault line) divides two of the square's four edges into two segments each that stand to one another in the ratio \( \rho \). The internal, coincident short edge of the medium rectangle and long edge of the small rectangle divides one of the square's other, two edges into two segments that stand to one another in the ratio \( \rho^4 \). Wikipedia

The  term "silver constant" should not be confused with the "silver ratio" of 1 + √2, which is one of the ratios from the family of metallic means. I posted about metallic means on Sunday, 3rd of March 2019. 

on January 22nd 2021

The changes were cosmetic and largely LaTeX-related, focused on improving the appearance of the cube root
symbol by raising the position of the 3 from its original default position using \leftroot {} and \uproot{}