The *Ishango bone* which has tally marks grouped in such a way that they appear to have been used for calculations is an example of what is called discrete mathematics and deals specifically with quantities that can be counted, now while this may seem confusing at first, you will understand the difference between discrete and continuous mathematics as we go on.

There are two sides to the coin of mathematics as previously stated, discrete and continuous mathematics, discrete in this context refers to mathematics that can be measured in terms of quantity and continuous refers to phenomenon such as clay, beer and land which are very difficult to calculate in finite terms. Continuous mathematics is a way of making the uncountable countable and the roots of geometry are deeply embedded in measurement.

While early civilizations found ways to deal with continuous quantities such as olive oil or wine, the origins of the word geometry lie with the farmers of the Nile Delta. Strangely it was out of the annual flooding of this very Nile Delta that this precise and sometimes necessary system of measurement was devised. You see, when the Nile flooded it washed away records of all those who owned pieces of land leading to the brightest of the time to devise ways to accurately mark out plots of land and geometry which literally translates to ‘*earth measure*‘ had developed.

Many of the figures now referred to by some as the fathers of geometry will be familiar to you if you have been reading this blog, some of those names include Euclid, Pythagoras and Thales of Ionia. When asked to name famous Greek mathematicians most people would name these very people and although we do not hear much of it in our modern schooling system the term *Euclid’s Elements* would have probably sounded quite familiar to our grandparents. As I have stated in a previous blog Euclid is often referred to as the father of geometry but as we delve deeper, that title more befits *Thales of Ionia* who made a study of geometry some 300 years before Euclid did.

it would be wise to bear in mind that there are no physical accounts of these so called theories that *Thales of Ionia* had conducted however there are many stories being passed around the campfire, so to speak, and one rather famous in nature states that he found a method for calculating the *Great Pyramid of Cheops* both around 2600 BCE.

At the time of writing this and utilizing resources that are available to me as current moment it is said that is not known specifically how the Egyptians constructed the pyramids more specifically what elements of geometry they utilized however tales insight into this brought us closer to understanding how complex these systems must have been.

## So How Exactly Did He Conduct The Experiment ?

Well, he waited until the sun was positioned in the sky such that his own shadow equaled his height, he then measured the length of the pyramids shadow from its base adding half the length of the base of the pyramid to the length of the shadow, by this reasoning Thales would give height to the pyramid.

Thales did not stop there though as he is also credited with such insights as the fact that the diameter of a circle always cuts the circle exactly in half or the observation that in an isosceles triangle, which for those of you who are not familiar with geometry is a triangle with two equal sides, the angles opposite the equal sides are also equal. These however should not come as a shock to anyone who is at all familiar with mathematics in the common era but put yourself for a second back in Thales time and imagine that someone with this level of intellect where to make these conclusions in your day, such observations where major steps forward in mathematics.

Thales through his observations does shift attention from the measurement aspect of geometry to the study of water, referred to as in variants, for properties of circles or isosceles triangles that remain constant irrespective of their size. The diameters of circles change but they all remain invariant. In bisecting the circle this deductive mindset provided a new way of thinking about mathematics moving away from mathematics being a purely practical enterprise that dealt with particular circles or triangles to the abstract study of generalities. It can be argued that they’ll set in motion the style of thinking from which modern mathematics has grown, and if there is one thing that can be regarded as having linked the disparate branches of geometry, it is the study of invariants.

## So Let’s Talk About Symmetry And Invariants

You will be probably familiar with symmetry as most normal people use it in the everyday informal sense of images that are pleasingly balanced for example a butterfly’s wings, or perhaps a 5 point petal design. These informal almost intuitive senses of symmetry are more formally developed in Euclidean geometry as the study of reflective and rotational symmetry.

if you are having difficulty understanding this, a butterfly has reflective symmetry much in the same way the word mum spelled ‘MUM’ has reflective symmetry, strangely enough psychologists have determined that human beings are far more drawn to human faces that are not symmetrical or asymmetrical than to those that are which are.

Just as well since most faces are less than symmetrical. 😉