The Hexagon and the Hexagram

Six is the number the circle counts by itself. Step the radius around the circumference and it fits exactly 6 times; join the 6 marks and the hexagon appears, the only regular polygon whose side equals its radius. 6 equal circles close around a 7th without a gap, the bee builds in hexagons, and 2 triangles laid across one another make the hexagram, the figure of the Hexad: 2 times 3, the material joined to the spiritual.

I. The Figure

Τὸ Σχῆμα

Euclid inscribes the regular hexagon at IV.15 and adds a corollary that no other polygon can claim: the side of the hexagon equals the radius of the circle.¹ The reason is in Layer 3. Join the centre to 2 adjacent corners and the triangle is equilateral, because its angle at the centre is a sixth of 360°, which is 60°, and the 2 radii are equal. Six such triangles tile the hexagon, and 6 is the number of them because 6 × 60° = 360°. Each interior angle of the hexagon is 120°, 2 of the triangle's.

The hexagon is one of exactly 3 regular polygons that fill the plane with copies of themselves. Aristotle states it in passing, as a thing agreed: "there are only three plane figures which can fill a space, the triangle, the square, and the hexagon".² Of the 3 the hexagon encloses the most area for a given perimeter, and Pappus, in the 4th century CE, credits the bees with knowing it: they chose the hexagon over the square and the triangle because it would hold more honey for the same expenditure of material.³

The hexagram is the hexagon's star, {6/2} in the notation of star polygons, and it's the one star in this study that falls into separate pieces: every second point of the hexagon makes a triangle, and the 2 triangles never meet along a single line. One points up, the other down; they cross at 6 points and enclose a small hexagon. The Hexad is 2 × 3 in the Temple's numerology, marriage through multiplication rather than through the Pentad's addition, and the hexagram draws the product: 2 figures of 3 points, multiplied across one another.⁴

II. The Construction

Ἡ Κατασκευή

1. Draw a circle with centre O and radius r. Mark any point A on it.
2. Keep the compass at r. With centre A, cut the circle at B; with centre B, cut it at C; continue around. The sixth cut returns exactly to A.
3. Join A, B, C, D, E, F in order: the regular hexagon. Join A, C, E and B, D, F: the hexagram.
4. Now draw the full circle of radius r about each of the 6 points. Six circles close around the first, each passing through the centre O and through its 2 neighbours' centres. This is the figure of 7 circles, the seed of the Flower of Life in Layer 13.

The 7-circle figure is the hexagon's generating pattern. Its 6 outer centres are the hexagon's corners, its 6 petals are the vesicae of Layer 2, and its 12 outer crossings are the corners of a larger hexagon turned through 30°. Every circle of 1 radius whose centre lies on another's circumference belongs to this lattice, which is why the Flower of Life grows from it without a new decision.

III. Number and Form

Ἀριθμὸς καὶ Σχῆμα

Six is the first perfect number: 1 + 2 + 3 = 6, the sum of its own divisors, as Euclid defines perfection in Book VII and as the Temple's numerology draws it in rows of 1, 2 and 3.⁵ It's also the first number that's both triangular, 1 + 2 + 3, and a product of 2 primes, 2 × 3. The hexagonal numbers, pebbles laid in nested hexagons sharing a corner, run 1, 6, 15, 28, 45, each n(2n − 1), and every one of them is also a triangular number, the 1st, 3rd, 5th, 7th and 9th.⁶

Six is the number of the cube's faces, and the hexagon is the cube's shadow. Look at a cube along its long diagonal, from corner to opposite corner, and its outline is a regular hexagon; its 9 visible edges draw the hexagon's 6 sides and 3 spokes meeting at the centre, half of the 6 spokes in the figure above. The octahedron, the cube's dual, shows the same outline from the same direction. The Hexad's "macrocosmic/ microcosmic" in the Temple's core meanings has a geometric form: the solid's projection into the plane, the greater whole seen in the lesser.⁴

IV. In the Ancient Witnesses

Παρὰ τοῖς Ἀρχαίοις

The hexagram belongs to no one people. Six-pointed stars are found on objects of the Bronze Age from Mesopotamia to Britain; in the Second Temple period the sign was used by Jews and non-Jews alike as a decoration, not as a specifically Jewish symbol; in the Middle Ages it served as a magical seal under the name of Solomon among Jews, Christians and Muslims.⁷ Its official use by a Jewish community begins in Prague, where Charles IV granted the community a flag in 1354, and its wide diffusion as the emblem of Judaism belongs to the 19th century.⁷ Gershom Scholem, the historian of Kabbalah, set out this history in 1949 under the title "The Curious History of the Six-Pointed Star".⁸ The figure was universal geometry for 3,000 years before it became a national sign; the Temple reads it as geometry.

In India the same figure is the ṣaṭkoṇa, the "six-angled figure", which the Sanskrit lexicon also records as a name of Indra's thunderbolt; in the tantric yantras it's the 2 superimposed triangles of the Goddess, the upward of Śiva and the downward of Śakti.⁹ Indra's thunderbolt is Zeus's own weapon in another tongue, and the sign that carries it is 2 triangles in the hand of the Sky-Father.

V. The Zevist Reading

Ἡ Ζευϊστικὴ Ἀνάγνωσις

The 2 triangles are the Pentad's 2 orientations held at once. The upward triangle is ascent, the soul's aspiration; the downward triangle is reception, power entering through the crown; the hexagram is the complete circulation, both movements present in 1 figure, which the Temple's reading of the star names as the whole of spiritual life.¹⁰ The Hexad's core meanings, material and spiritual, macrocosmic and microcosmic, are the 2 triangles' names, and their crossing is the Temple's living harmony: each contributes, and the whole gives their union a complete expression.⁴

The hexagon is the figure of sacred service. Six is priesthood, cleanliness and habituation in the Temple's numerology, and the bee's cell is the right image: a form repeated without variation, each cell exact, the whole comb stronger than any wall of another shape.⁴ The care of a sanctuary, the preparation of its objects, the steady observance of its duties make devotion inhabitable, and the hexagonal tiling is devotion inhabited: no gap, no overlap, no cell out of measure. Ma'at in a sanctuary looks like a honeycomb.

The 7 circles add the centre that the 6 serve. The Hexad's perfection, the Temple notes, is the full participation of its parts; the seventh circle is the whole in which they participate, and Layer 7 takes it up. The Hexad is also the number of the 6 directions in space, up and down added to the 4 quarters of the cross, and the hexagram's 3 axes, drawn through its opposite points, are the 3 axes of the cube and the octahedron that Layer 12 will raise. The Hexad is the Tetrad's boundary extended into height and depth.

VI. Measure

Τὸ Μέτρον

VII. Sources and Reading

Πηγαί

1. Euclid, Elements IV.15 with its corollary. Joyce.
2. Aristotle, On the Heavens III.8, 306b5 to 7. English (Stocks), Part 8 · Greek, Scaife Viewer.
3. Pappus of Alexandria, Collection V, preface ("On the Sagacity of Bees"), in Ivor Thomas, Selections Illustrating the History of Greek Mathematics II (Loeb, 1941), pp. 591 to 593. MacTutor, Pappus (quotes the passage) · Thomas, vol. II, archive.org.
4. Temple of Zeus, 6 · The Hexad.
5. Euclid, Elements VII, Definition 22. Joyce.
6. Eric W. Weisstein, "Hexagonal Number", MathWorld. mathworld.wolfram.com.
7. Gershom Scholem, "Magen David", Encyclopaedia Judaica, reproduced by the Jewish Virtual Library. jewishvirtuallibrary.org.
8. Gershom Scholem, "The Curious History of the Six-Pointed Star: How the 'Magen David' Became the Jewish Symbol", Commentary, September 1949; revised as "The Star of David: History of a Symbol" in The Messianic Idea in Judaism (Schocken, 1971). commentary.org.
9. Monier-Williams and other Sanskrit lexica, ṣaṭkoṇa: wisdomlib.org. Arthur Avalon (John Woodroffe), Hymns to the Goddess (1913), Tārāṣṭakam, note on the ṣaṭkoṇa: sacred-texts.com. Madhu Khanna, Yantra: The Tantric Symbol of Cosmic Unity (Thames and Hudson, 1979).
10. Temple of Zeus, The Pythagorean Star and the Two Orientations.
11. Temple of Zeus, How to Use the Numbers in Magick & Meditation.