At the heart of Pythagorean philosophy is the triangular pattern of 10 dots or points called the 'tetractys'. Mathematicians regard it as the fourth of the so-called 'triangular numbers.' These are numbers that are the sums of triangular arrays of dots, each denoting the number 1. For example, 1, 1+2=3, 1+2+3=6 and 1+2+3+4=10 are the first four triangular numbers. The tetractys, however, meant far more than this to the followers of Pythagoras. In fact, so much did they honour it that they swore to their teacher, who had recognised the meaning and importance of this pattern, an oath of loyalty that mentioned the symbol as his discovery. Just as the number 1 is the source of all number, so the Monad symbolized by a point or dot is the divine origin of all phenomena. Integers increase until they attain their perfect completion in the number 10, the Decad. The tetractys is more than just a representation of this number. It symbolizes the 10-fold nature of Divine Unity as it manifests in existence — physical and superphysical. As such, it is the template for constructing holistic objects that possess sacred geometry. By constructing them from tetractyses, objects are revealed to express numbers — the numbers of dots needed to assemble them. When objects possess 'sacred geometry' — not the vacuous label found in many books on the subject but the genuine version, these numbers acquire cosmic, rather than mere human cultural, significance. As such, some of them are relevant to science, as this website will demonstrate. It was this amazing power to reveal certain numbers of universal significance, as well as the mathematical nature of the divine design of nature, that made the tetractys so valuable and sacred to the ancient Pythagoreans. That amazing power will be illustrated for the first time in this website.

The tetractys is a set of four horizontal rows of 1, 2, 3 & 4 dots. The properties of objects possessing sacred geometry can be expressed naturally in terms of these four integers. Any dot which is part of a tetractys will be called a 'yod' because this is the name of the tenth letter (י) of the Hebrew alphabet, which is shaped somewhat like a dot. Such mingling of Hebrew and Pythagorean concepts is not the author's invention. As much as ultra-orthodox Jews may deny it, Pythagorean mathematics influenced early European Kabbalists, who noticed the analogy between the 10 Sephiroth of the Etz Hayim (Tree of Life) and the 10 points of the tetractys.

The 10 yods of the tetractys shown opposite consist of the three black yods forming its corners, i.e., marking its shape in a minimal way, and seven red yods, six of which are the corners of a hexagon (assuming the tetractys is equilateral) and the seventh is its centre. These seven yods will be called 'hexagonal yods.' This 3:7 division expresses the differentiation of the 10 degrees of freedom symbolised by the 10 yods into three that are pre-formative and seven that are formative, determining the physical manifestation of holistic phenomena. It is important to realise that the seven yods will be called 'hexagonal yods' irrespective of whether the tetractys is actually equilateral. Even if the triangle in a tetractys array is isosceles or scalene, in which case they are not the centres and corners of a hexagon, these seven yods will still be referred to as 'hexagonal yods.'