Self-portrait by Dürer when aged 26.
Albrecht Dürer was born in Nuremberg in 1471, third of a family of eighteen children. Were he still living, he would be celebrating his 549th birthday today. Dürer’s artistic genius was clear from an early age, as evidenced by a self-portrait he painted when just thirteen [TM187; or search for “thatsmaths” at irishtimes.com ].
In 1494, Dürer visited Italy, where he travelled for a year. A novel connection between art and mathematics was emerging around that time. By using rules of perspective, artists could represent objects in three-dimensional space on a plane canvas with striking realism. Dürer was convinced that “the new art must be based upon science; in particular, upon mathematics, as the most exact, logical, and graphically constructive of the sciences”.
Continue reading ‘Changing the way that we look at the world’
For many decades, a search has been under way to find a theory of everything, that accounts for all the fundamental physical forces, including gravity. The dictum “physics is geometry” is a guiding principle of modern theoretical physics. Einstein’s General Theory of Relativity, which emerged just one hundred years ago, is a crowning example of this synergy. He showed how matter distorts the geometry of space and this geometry determines the motion of matter. The central idea is encapsulated in an epigram of John A Wheeler:
From cheetahs chasing gazelles, through coastguards saving shipwrecked sailors, to missiles launched at enemy aircraft, strategies of pursuit and evasion play a role in many areas of life (and death). From pre-historic times we have been solving such pursuit problems. The survival of our early ancestors depended on their ability to acquire food. This involved chasing and killing animals, and success depended on an understanding of relative speeds and optimal pursuit paths.
characterizes the intrinsic geometry of a surface.
It was a great shock to the Pythagoreans to discover that the diagonal of a unit square could not be expressed as a ratio of whole numbers. This discovery represented a fundamental fracture between the mathematical domains of Arithmetic and Geometry: since the Greeks recognized only whole numbers and ratios of whole numbers, the result meant that there was no number to describe the diagonal of a unit square.
where 
is the radius. Some of us may also remember that the volume of a ball — the interior of a sphere — is 
.
, the length of the major axis. The gardener puts down two stakes and loops a piece of rope around them. Using a stick, he pulls the loop taut, marking the points around a curve. This is illustrated here.
Ireland has four provinces, the principal city in each being the provincial capital: Belfast, Cork, Dublin and Galway. The map here shows the location of these cities. Now imagine a company that needs to visit and to deliver goods frequently to all four cities. Where might they locate their HQ to minimize transport costs and travel times?
is the position of the HQ and 
are the positions of the cities.
Every circle has the property that the distance around it is just over three times the distance across. This has been known since the earliest times [see 
for 
is rotated in 3-space about the x-axis.
![Mercator projection of the Earth, truncated at 75 degrees North and South [Wikimedia Commons, author: Strebe].](https://thatsmaths.files.wordpress.com/2015/05/mercator-projection-75deg.jpg)
![From left to right: Figure-of-8 knot 4_1, knot 6_3 , knot 7_1, and knot 8_5 [Fig from Van Wijk (2006)].](https://thatsmaths.files.wordpress.com/2015/01/knots-vanwijk.jpg)
so that it generates a plane curve as it moves. This, a curve results from a mapping ![{\mathbf{\gamma} : [a,b]\longrightarrow \mathbb{R}^2}](https://s0.wp.com/latex.php?latex=%7B%5Cmathbf%7B%5Cgamma%7D+%3A+%5Ba%2Cb%5D%5Clongrightarrow+%5Cmathbb%7BR%7D%5E2%7D&bg=ffffff&fg=000000&s=0 "{\mathbf{\gamma} : [a,b]\longrightarrow \mathbb{R}^2}")
. 