Now mostly recognised through the mythology of witchcraft movies, graffiti and gutter-press Satanist exposés, the pentagram has almost everywhere become disembodied from its roots in geometry - everywhere, that is, except in Freemasonry. There we see it on the doorstep to London’s Freemasons’ Hall, for example, and we see its radiant stellar properties in the Royal Arch Degree ritual. Fully developed only in the 19th century, the ritual is in part based on the Platonic system wherein five solid geometrical bodies embody the principles on which the universe is formed. The candidate must answer three questions on the nature and origin of Platonic bodies and their representations within the ritual.
    Translated into the ritual, Plato teaches that the four elements of earth, fire, air and water were formed from atoms and presented respectively as the cube, the tetrahedron, the octahedron and the icosahedron.
    The 12-sided dodecahedron (made up of pentagons) represents the ultimate fifth element, by the use of which the Great Geometrician completes the universal creation. Plato received this doctrine from the followers of Pythagoras (born at Samos in c.596BC) who formed a brotherhood bound by oath not to reveal the teachings of the school. Pythagoras had taught his followers that the universe was explicable in mathematical terms, but given the scope of this idea it is not surprising that the mathematical symbols were also immersed in philosophical meanings. Plutarch (c.50-120 AD) recorded the Pythagorean belief that the earth was made from the regular hexahedron, fire from the pyramid, air from the octahedron, water from the icosahedron and the heavenly sphere from the dodecahedron. It was natural then for Jews, Christians and pagans to see meaning in this twelve-fold symbolism : 12 Tribes of Israel, 12 Apostles, and the 12 signs of the Zodiac, for example.
    In the dodecahedron, we see a relation between the divine number Five and the heavenly Twelve, for this body is made of regular pentagons. The pentagon is constructed from two lines which are in a Golden Section proportion to each other. This is sometimes called the Golden Mean, or the Divine Proportion, being the relation between two lengths, first described in Euclid’s Elements, Book VI, Proposition No 30 (to cut a given finite line in extreme and mean ratio). The Golden Mean is the proportion of a divided line such that the smaller part is in the same relation to the larger part as the larger part is to the whole.
    In other words, a is to b in the same relation as b is to l. If we now use the two lengths a and b to construct a triangle, we arrive at the basis for constructing a pentagon.
    The pentagon is further constructed by using this triangle and making a circle to contain it.
    If we now transfer the smallest side (the base) of the triangle around the circle in consecutive order, we now see a regular pentagon within the circle.
    Merely in following this process of construction yourself, your mind starts to contemplate a surprising order and consistency that does not seem merely incidental.
    The five-sided figure generates not only its double number of ten triangles, together with a proportionally reduced inverted reproduction of itself heading into the centre ad infinitum, but with the application of a protractor we find not only that each point on the circle occurs every 72 degrees on the 360 degree circle (as we would of course expect from a circle divided by 5) - but that all of the internal degrees are either 36, its double, 72, or its triple 108. Lovers of numerology will note that like the 360 degrees of the circle, each of these numbers reduces to the odd number of 9 (3+6; 7+2; 1+8), each is divisible by 9, and all are made from an odd and an even number. An imaginative observer of these geometrical phenomena might wonder if he or she hadn’t stumbled on one of the Great Geometrician’s fundamental building-principles! And this is what happened, so that the figure attained a magical power - and before the 17th century (and for some after it), magic and geometry were inseparable : the one who knew what to do with the geometry (viz : building) had magical powers because he held the divine secrets in the palm of his hand. Look at the illustration from Agrippa’s Three Books of Occult Philosophy (English version, 1651) where we see Man holding aloft two pentagrams, his arms like dividers, with the plumbline at his solar-plexus and the upright builders’ square at his feet!
    The pentagram was seen as embodying a divine proportion which was itself written into the form of Man in the Divine Archetype (made in His Image).
    Agrippa writes : “The number five is of no small force, for it consists of the first even, and the first odd, as of a Female, and Male. For an odd number is the Male, and the even the Female.” The Pythagoreans called the five odd numbers (including 1) the gnomones; the gnomon was “one who knows or examines” in Greek, as well as the word for the index of the sundial (gno- is the root of words relating to knowledge and intelligent discernment, such as gnosis).
    Apart from these and many other speculative deductions, there is a key geometrical reason for the fugure’s importance in thinking about the elements of creation, namely, the idea of incommensurability. Briefly, incommensurability is that phenomenon wherein two magnitudes cannot be measured by the same unit, but which can nonetheless be geometrically constructed. This idea is most elegantly incorporated in the grand design of the pentagram and the dodecahedron. The mathematician and astronomer Kepler (1571-1630) was reported to have said that the two treasures of geometry were Pythagoras’ Theorem and the division of a line into the mean and extreme ratio. The most perfect of all incommensurable ratios is the Golden Mean. Pythagoreans found that this ratio is most easily to be found in a pentagram, where the sides of the triangle are all in this relation to each other.
    This property of the figure and its easy replication to an indefinite number of similar figures led them to believe in its magic powers which are related to growth and creation : the essential principles for a magical talisman. Pythagoreans denoted the five angles by the letters UGIQA : HYGITHA, from the Greek verb hygiazo, meaning to make sound, healthy, or to cure. An interesting variant on this idea can be seen on the seal of Sir Robert Moray, first President of the Royal Society, advisor to Charles II, friend of Elias Ashmole - and a Freemason initiated into an operative Edinburgh lodge on 20 May 1641.
    Around his personal pentagram Moray put the Greek letters whose English equivalents are AGAPA, from the Greek verb agapao, which means to give affection, to love, thus showing Moray’s powerful belief in the talismanic power of loving friendship and the joys of fraternity.
    We are just beginning to see again that the roots, not only of architecture, but of higher religion, chime in with geometrical discovery : the creating Mind of the Grand Geometer extended. Nevertheless, one should consider Blake’s statement : “God is not a spiritual diagram.” The geometry comes from the Mind, not the Mind from the geometry. The pentagram embodies this realisation; it is a gift from the past to the future.