Euclid

Pascal’s Triangle

One of the most interesting Number Patterns is Pascal’s Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher). …

Diagonals. …

Symmetrical. …

Horizontal Sums. …

Exponents of 11. …

The same thing happens with 11⁶ etc.

Squares. …

Fibonacci Sequence.

The Coefficients of the Binomia Theorem from Pascal’s Triangle

Pascal’s triangle formula is (n+1)C(r) = (n)C(r – 1) + (n)C(r). It means that the number of ways to choose r items out of a total of n + 1 items is the same as adding the number of ways to choose r – 1 items out of a total of n items and the number of ways to choose r items out of a total of n items.

The Fibonacci sequence with Pythagorean triples

The sum of the squares of consecutive Fibonacci numbers is another Fibonacci number. Specifically we have the following right triangle. The hypotenuse will always be irrational because the only Fibonacci numbers that are squares are 1 and 144, and 144 is the 12th Fibonacci number.

Pascal’s triangle is commonly used in probability theory, combinatorics, and algebra. In general, we can use Pascal’s triangle to find the coefficients of binomial expansion, the probability of heads and tails in a coin toss, the probability of certain combinations of things, and so on…

Euclid

Postulate 1 ‘[It is possible] to draw a straight line from any point to any point’.

Postulate 2 ‘[It is possible] to produce a finite straight line continuously in a straight line’.

Postulate 3 ‘[It is possible] to describe a circle with any centre and diameter’.

Geometry

Euclid’s theorems are still true and his methods are still admired. For millenia his books have been studied and referenced, though they are no longer used as a school text-book.53 He entitled his principal work Elements, and it was intended to be a foundational work in the subject, a starting point. The same Greek word (stoikheia) also means the letters in the alphabet, and Euclid’s elements are to geometry what letters are to language: the building blocks or basic components.

One of the most outstanding features of Euclid’s work is its structure: the first book contains a number of definitions, postulates and common notions, and the following twelve books endeavour to introduce or assume no extraneous material as they progress, but only to construct from definitions and propositions already done. Thus, for any proposition one can trace back the reasoning for a particular result through earlier propositions until one comes back ultimately to the original postulates and common notions.

This trace can be illustrated by drawing a proof tree, of which an example is given below in Figure 3, to illustrate the reasoning for Pythagoras’ Theorem. Of course Euclid was not infallible, and there are occasionally holes in the arguments, but these should not be allowed to detract from the overall aim and success of his method. Another outstanding feature is the thoroughness with which propositions are proved, as will become apparent in the example given below. Let us first review the Elements.

Book 1 builds from twenty-three definitions, five postulates, and nine common notions.54 The definitions explain the basic terms of geometry, what is meant by words such as ‘point’ or ‘line’. The common notions are axioms or self-evident truths; statements that any sensible person would take as true, although it is not possible to prove them. For example, Common Notion 1 is ‘Things which are equal to the same thing are also equal to one another’. The postulates are unproved assertions about geometry. The first three postulates are assertions that amount to the possibility of doing geometry.

Postulate 1 ‘[It is possible] to draw a straight line from any point to any point’.

Postulate 2 ‘[It is possible] to produce a finite straight line continuously in a straight line’.

Postulate 3 ‘[It is possible] to describe a circle with any centre and diameter’.

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