# The Trinity Mathematical Society, Cambridge

### Founded in 1919

**The Trinity Mathematical Society**was founded in 1919 by a group of Cambridge undergraduates, supported by G H Hardy. Its aim, from the time it was founded, was to:-

... promote the discussion of subjects of mathematical interest.We give a few of the Society's rules below (note the beautiful mathematical Rule 4) [3]:-

1. The Society shall be called the Trinity Mathematical Society.There are two symbols associated with the Society. One is an apple, which clearly is a reference to Isaac Newton who was a student at Trinity College and later a fellow of the College. The standing orders of the Society state [3]:-

2. The object of the Society shall be the promotion of activities pertaining to the Mathematical Sciences, and the furtherance of social intercourse among its Members.

3. All members of Trinity College, and no others, may apply for Ordinary Membership. Members of other Colleges, and no others, may apply for Associate Membership. Senior members of the University and (in special cases) others may be invited to Honorary Membership.

4. The affairs of the Society shall be managed by a Committee consisting of at least three Officers, who shall be Ordinary Members of the Society: a President, a Vice-President, a Secretary, and up to three others. The roles of the Committee shall be detailed in the Standing Orders. The partial map from Officers to Ordinary Members shall be injective.

5. The Society shall hold at least two meetings in each Full Term except in the Easter Term. Business involving the election of Officers or the alteration of these Rules must be transacted at a meeting, and may not be transacted outside of Full Term.

Meetings shall be declared social by the dropping of an apple. The Society tie, if it exists, shall be "Azure, a ripe apple (with stalk and leaf) proper, within a triangle or".The President of the Society also bowls an apple as the first ball in the annual cricket match against the Adams Society of St John's College, Cambridge.

The second symbol, which is the official logo of the Society, is the unique smallest simple squared square. It is a square divided into 21 squares all of different sizes. That it is simple means that no proper subset of the squares of size at least 2 forms a rectangle and smallest in that no square can be so divided with fewer squares. This squared square was discovered in March 1978 by A J W Duijvestijn using a computer search and published in his paper

*Simple perfect squared square of lowest order*(1978) which contained the order 21 simple perfect squared square (see [2]). Let us look at why the Society has this particular logo.

Henry Dudeney produced a puzzle in 1902 which asked for a square to be dissected into squares, but there was a rectangle in this solution. Several mathematicians examined the problem of dissecting squares into squares including Max Dehn in 1903. Nikolai Luzin, in 1930, conjectured that it was impossible to dissect a square into a finite number of squares all of different sizes. The problem was solved by four members of the Trinity Mathematical Society, namely Rowland Leonard Brooks (1916-1993), Cedric Austin Bardell Smith (1917-2002), Arthur Harold Stone (1916-2000) and William Thomas Tutte (who has a biography in this archive). They all met in 1936 as follows [5]:-

Cedric Smith introduced Leonard Brooks to Arthur Stone and, to reciprocate, Leonard Brooks introduced all of us to a chess-playing friend, a chemist William Tutte.Arthur Stone had seen Henry Dudeney's puzzle and thought that Dudeney was implying that dissecting a square into a finite number of squares all of different sizes was impossible. He tried to prove this but failed. However he managed to dissect a 176 by 177 rectangle into 11 unequal squares. After much work on dissecting rectangles into squares, the team of four eventually solved the problem of finding a square which could be dissected into 39 smaller unequal squares. They improved this, first to one with 28 unequal squares, then to one with 26. However, these solutions were not simple in the sense defined above. Rowland Brooks, Cedric Smith, Arthur Stone and Bill Tutte published their results in the paper

*The dissection of rectangles into squares*(1940), see [1]. Ten years later Tutte found a simple squared square with 69 different size squares in the dissection which he published in

*Squaring the square*(1950), see [4]. Let us note that Rowland Brooks and Cedric Smith were successive Presidents of the Trinity Mathematical Society, Brooks in 1937 and Smith in 1938.

The Standing Orders of the Society state [3]:-

... the logo of the Society shall be the (unique) smallest simple squared square, with the largest partitioning square in the top left corner and the largest of the squares adjacent to this to its right (rather than below it).

The 'best' squared square result from Rowland Brooks, Cedric Smith, Arthur Stone and Bill Tutte's 1940 paper was a square dissected into 26 unequal squares and can be seen at THIS LINK.

This was made as a table top in December 1982 by R R G Rivington and the table is now kept in Trinity College and a picture of the table is at THIS LINK.

The smallest possible solution was discovered in 1978 and so the present logo of the Society is at THIS LINK. You can see more about it at the Society's web page

Rowland Brooks, Cedric Smith, Arthur Stone and Bill Tutte are in the 1938 photograph of the Trinity Mathematical Society. We note that Hermann Bondi, Abram Besicovitch and Charles Coulson are also in this photograph. The 1938 photograph of the Trinity Mathematical Society can he seen at THIS LINK.

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The Trinity Mathematical Society states [3]:-

Today the Society has over 800 members and each year holds an extensive range of talks and social events for mathematicians and interested non-mathematicians from Trinity and other colleges. Our talks range from discussions of interesting mathematics not covered by the Tripos to light-hearted expositions of new, but accessible, research. We believe the Trinity Mathematical Society is the oldest extant subject society at any university in the country. The Trinity Mathematical Society is affiliated to the Archimedeans, the Cambridge University mathematics society. As part of this affiliation, we admit members of other colleges' mathematics societies (currently only the Adams Society) to our talks for free - and in return, members of the Trinity Mathematical Society are admitted to Adams Society talks for free.

Visit the society website.

### References (show)

- R L Brooks, C A B Smith, A H Stone and W T Tutte, The dissection of rectangles into squares,
*Duke Mathematical Journal***7**(1940), 312-340. - A J W Duijvestijn, Simple perfect squared square of lowest order,
*J. Combin. Theory Series B***25**(1978), 240-243. - The Trinity Mathematical Society website. https://tms.soc.srcf.net/about-the-tms/
- W T Tutte, Squaring the square,
*Canadian J. Math.***2**(1950), 197-209. - W T Tutte, Squaring the square, in
*Mathematical games, Scientific American*(November 1958).

Last Updated February 2018