# Fermat Point by Suman Vaze

The maths behind Issue 02’s cover

Fermat Point by Suman Vaze

Suman Vaze sits on her small balcony in crowded, bustling Hong Kong, with a view, just about, of a beautiful Chinese Banyan tree tenaciously growing on a steep stony slope, and paints mathematics. Inspired by the abstract expressionism of Rothko, the radical and influential work of Picasso, and the experimental models of Calder, she fully embodies Hardy’s belief that mathematicians are ‘maker[s] of patterns’. Our front cover is one of her pieces: the bold colours proclaim the eponymous Fermat Point – the point that minimises the total distance to each vertex of a triangle – along with its geometrical construction. Add an equilateral triangle to each side of the original triangle then draw a line connecting the new vertex of the equilateral triangle to the opposite vertex of the original: the intersection of these lines gives the Fermat point. Not only do these lines all have the same length, but the circumscribed circles of the three equilateral triangles will also intersect at the Fermat point.

The Octagonal Numbers are those of the form $3n^2 – 2n$. Taking the digital roots of the octagonal numbers gives the repeating sequence 1, 8, 3, 4, 2, 6, 7, 5, 9,…

All of her paintings combine beauty with a deeper mathematical meaning. Suman describes the rock gardens in Ryoanji Temple in Kyoto as being ‘enclosed in a rectangular courtyard surrounded by lush Japanese gardens. The rock garden within is an austere arrangement of rocks on neatly raked gravel. Their proportions and positions defy symmetry yet they have an aesthetic balance’ and they moved her to create The Ryoanji Suite. The Octagonal Numbers also reflect her love for sequences and patterns: ‘in numbers, shapes, operations, movements,…’

Much of her inspiration and, one assumes, her motivation comes from the students she teaches in Hong Kong. Through them, she has seen the wide range of emotions that one often experiences when confronted with mathematics: from horror and despair to gratifying relief and triumphant joy. Where a pupil will fall in this scale depends, she believes, on ‘their approach to the subject and their ability and willingness to immerse themselves in it. Those who enjoy the subject are the ones who are in a sense fearless with new ideas and are able to view it from different perspectives and hence develop greater intuition with the concepts.’

The Ryoanji Suite is constructed in a series of 20 strokes, made so that their intersections formed the consecutive numbers 1 to 9. The artwork was the process of producing the intersections in a particular order and the result of the process is the calligraphy, Ryoanji.

For Suman, art provides her with a way to express these concepts. Lady Ponzi (on the banner), for example, illustrates the dissection of an equilateral triangle into four pieces that can be reassembled to form a square. The problem was solved in 1902 by Henry Dudeney, an English author and mathematician who set puzzles for various English newspapers and magazines, with the solution having the interesting property that the pieces can be hinged to smoothly rotate between a triangle and a square. Dudeney, incidentally, is also credited for publishing the first crossnumber puzzle.

The wider audience, too, should be grateful for Suman’s ability, as her artwork is a window into the beautiful, intricate and magical world of mathematicians, makers of patterns.

Suman Vaze is an artist and mathematics educator at King George V School, Hong Kong. Her work has been exhibited in mathematics conferences all around the world, including the US, Canada, Hungary, the Netherlands and Korea, along with several solo shows in Hong Kong. For more of her art, go to sites.google.com/site/vazeart.

(Title image: Lady Ponzi by Suman Vaze)

Huda Ramli is a PhD student at UCL, working on stochastic models of atmospheric dispersion.

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