The second convex regular polygon in our sequence is the square {4}.

Intrinsic Properties

A square has four equal sides, and four equal angles each measuring 90°. It has two diagonals which bisect each other at right angles. Another way of saying this, is to say that a square is a special quadrilateral (four sided polygon) which is both equilateral and equiangular. It is the intersection of the family of (equiangular) rectangles which have equal diagonals, and the family of (equilateral) rhombuses which have perpendicular diagonals.


We shall describe the symmetries of the square, and this will set the pattern for the remaining regular polygons.

• Can you find all the transformations that map the square onto itself? How many are there? Does the pattern fit with the symmetries we found for the equilateral triangle?   


                                                            (a)                     (b)                   (c)                     (d)
                                                            (e)                     (f)                    (g)                     (h) 

Now as previously, make your transformations correspond to the eight images to the right of the initial left square.
Finally, check your results.

This group of eight distance preserving transformations is called the dihedral group, D4. In general, the symmetries of the regular n-gon form the dihedral group of degree n, Dn, containing 2n symmetries – n rotations (including the identity), and n reflections. You might notice that for the equilateral triangle we used the term symmetric group, S3. In fact, this coincides with D3, but for n = 3 the symmetries permute the vertices in all possible 3! ways; that is, we get all possible symmetries (of the points). This only happens for n = 3.

Two interesting results

As for the equilateral triangle, there are some interesting mathematical results in which the square unexpectedly appears. First a result due to Finsbury and Hadwiger which is a little reminiscent of the Napoleon Theorem for triangles.

On sides AB, BC of an arbitrary triangle ABC, construct squares ABUV and BCYX. Now consider the respective centres S, T of these squares, the midpoint M of AC, and the midpoint Z of XY. Can you make a conjecture?

Click to confirm your answer. Of course it is a square!

A more satisfying result, also an obvious analogue of Napoleon’s Theorem, is the following.

Draw any parallelogram, and construct a square outwards on each side. Consider the centres of these four squares. Can you make a conjecture? What previous result does this remind you of?

Click to confirm your answer. This is obviously an analogue to the Napoleon Theorem for triangles. The result holds true if the squares are constructed inwards on the sides of the parallelogram. If the squares are constructed on the sides of a general quadrilateral, the centres no longer form the vertices of a square, but a partial result still remains true. What is it? This is known as van Aubel’s Theorem.

You might like to think of further ways in which these results might be generalized.

Some square recreations

The title may make us think of chess, but there are other games and puzzles which rely on the square for their structure. The first of these are the well known magic squares.

You are given a 3 x 3 square, and are asked to place the digits 1, 2, ... , 9 in the lattice squares so that the rows, columns and diagonals all give the same sum. What is the sum? (It is always the same.) Can you show that a certain digit must occur in the centre square?

• Now try your hand at this 4 x 4 magic square, using the numbers 1, 2, ... , 16. What is the common sum? No central square to help out this time!

Now check your answers.

Another topic which may be classed as recreational is the topic of square dissections. It is obviously possible to subdivide a given square into smaller equal squares. But can we dissect a given square into a finite number of unequal squares? In 1978, Duijvestijn discovered a dissection into 21 smaller squares. It is known that this is the smallest possible number.

Duijvestijn’s dissection of a square into 21 smaller squares.

If you are looking for an easier challenge, you might try dissecting a square into acute angled scalene or isosceles triangles. (Each can be solved using not more than ten triangles.)

Some real life occurrences

To complete our survey of the square, let us consider where the square occurs in real life.

Make a list of at least half a dozen occurrences of the square in real life. How many can you find? This should be easier than the triangle!

Now click the photograph for some ideas.


The theory of groups : http://internal.maths.adelaide.edu.au/people/pscott/groups/gpf/

Finsler and Hadwiger result : Finsler, P. and Hadwiger, H., Einige Relationen im Dreieck, Commentarii Mathematici Helvetici 10 (1937) 316 – 326.

van Aubel’s Theorem : van Aubel, H. H., Note concernant les centres de carrés construits sur les côtés d'un polygon quelconque, Nouvelle Correspond. Math. 4 (1978) 40 – 44.

Magic Squares : http://mathforum.org/alejandre/magic.square.html

Square Dissections : Duijvestijn, A. J. W., A Simple Perfect Square of Lowest Order, Journal of Combinatorial Theory (Series B) 25, (1978) 240 – 243.

In the 3 x 3 case, the sum of the digits is 45, so each row/column/diagonal sum has to be 15. Now if the digits in the rows are a, b, c; d, e, f; g, h, j, then
90 = (d + e + f) + (b + e + g) + 2(a + e + j) +2(c + e + g)

= 6e + (a + b + c) + (c + f + j) + (j + h + g) + (g + d + a)

= 6e + 60, whence e = 5. The rest comes from trial and error!

In the 4 x 4 case, the sum is 34. You will notice many 34s in the given solution.

9 4
7 5 3
6 1 8
15 14 4
12 6 7 9
8 10 11 5
13 3 2 16

The eight transformations are:

(a) The identity map, which leaves the square unchanged.
(b) An anti-clockwise rotation of 90° about the centre.
(c)  An anti-clockwise rotation of 180° about the centre.
(d)  An anti-clockwise rotation of 270° about the centre.
(e) A reflection in the vertical mirror line.
(f) A reflection in the horizontal mirror line.
(g) A reflection in the positive diagonal mirror line.
(h)  A reflection in the negative diagonal mirror line.

These eight symmetries of the square form the dihedral group D4.