Polygons and star polygons in Freemasonry
Triangle, pentagram, hexagram and the blazing star are fixed elements of Masonic symbolism. Were they only signs, or was there knowledge behind them?
This page deals with the geometric figures and numbers of Freemasonry. What Freemasonry is, where it comes from and how it is structured is on the page Freemasonry: history, degrees and symbols.
Why is Freemasonry in the section on mysticism? Because it works with symbols in which number and figure play a large part. Several streams flow together in it: the customs of the stonemasons (see The signatures of the cathedral builders), the learning of the Renaissance with Hermeticism, Kabbalah and Rosicrucianism, and the ideals of the Enlightenment. How large a share the esoteric currents had in early Freemasonry is disputed among historians. Some lodges are even named after Pythagoras.
The figures
Freemasonry works with symbols in which numbers and geometric figures play a large part. In publications, ritual objects and lodge rooms polygons and star polygons appear again and again. Their meanings are not strictly fixed:
- The triangle can mean the “Great Architect of the Universe”, but also the degree of Entered Apprentice.
- The pentagram can mean the human being, but also the degree of Fellow Craft.
- The hexagram can mean the union of God and world, or polarity as such, but also the degree of Master Mason.
The four start figures of the tetractys in particular, triangle, hexagram, pentagram and double pentagram (which might be read as the “blazing star”), are among the most important symbols. They also appear on the jewels, the insignia lodge members wear during ritual work. Besides these four there are many other figures in Masonic symbolism, but almost always symmetries around a centre.
The four start figures of the tetractys: triangle, hexagram, pentagram and double pentagram. The Masonic 'blazing star' is usually drawn as a simple five-pointed star.
Square and compasses
The emblem of Freemasonry is the square and compasses, interlocked, sometimes with the letter G in the middle. Freemasons themselves often read this G as geometry, which is exactly what one practises with compasses and square. Both tools can be linked with many pairs of concepts, in the broadest sense like yin and yang. Could it not be that the spiritual forerunners of Freemasonry, such as Pythagoras, and later “initiates of geometry” discovered the secrets of creation with precisely these tools? With compasses and square one draws circles, polygons and star polygons, the very figures explained number-theoretically under The key to the tetractys. Here are connections where one need no longer be content with mystical interpretation. They could give Masonic symbolism something like a scientific foundation.
Why were these figures so important to the founding fathers? Were they only symbols? Discretion is a virtue of the brotherhood, and what the founders themselves thought they seldom wrote down. Whether more than a language of images stood behind the figures we do not know.
Point, line, surface and solid in Samuel Prichard
In 1730 Masonry Dissected by Samuel Prichard appeared in London. It made Masonic custom public in detail for the first time, went through many editions and translations and prompted the Grand Lodge of London to change its rituals. It is built as a question-and-answer piece, and in it point, line, surface and solid appear quite directly, the four stages that are also read as the tetractys. Prichard does not mention Pythagoras. Its very first sentence names the liberal arts as the foundation of Masonry, above all the fifth, geometry. The Entered Apprentice’s catechism reads:
Q. How many Principles are there in Masonry?
A. Four.
Q. What are they?
A. Point, Line, Superficies and Solid.
Q. Explain them.
A. Point the Centre (round which the Master cannot err) Line Length without Breadth, Superficies Length and Breadth, Solid comprehends the whole.
Remarkable is the image in the last answer: the point is “the Centre round which the Master cannot err”. What is meant is probably the compasses. Whoever sets their point firmly in the centre draws a circle that deviates nowhere. Later lectures made of this the image of the point within a circle: the circle shows the boundary within which one should keep one’s conduct. Notable too is that the line is explained here almost word for word as in Euclid, as “Length without Breadth” (see From point to solid).
And in the Fellow-Craft’s part:
Q. Why was you made a Fellow-Craft?
A. For the sake of the Letter G.
Q. What does that G denote?
A. Geometry, or the fifth Science.
A clear sign that Pythagorean-Platonic thought ranked high at least in the early days of Freemasonry. Early Masonic literature also reproduced these passages faithfully. The full text is freely available at archive.org.
The tetractys in the Rebis
The woodcut from Heinrich Nollius’ Theoria philosophiae hermeticae (Hanau 1617) shows a Rosicrucian allegory, the “philosophical egg”. Nollius probably did not invent the image, it already appears in an edition of the Azoth of 1613 attributed to Basil Valentine. Rebis, Latin for “two things”, denotes in alchemy the union of two principles into a higher one, such as sulphur and mercury, but above all of the male and female principle in the hermaphrodite in the preparation of the philosophers’ stone.
For Freemasonry it is remarkable that the hermaphrodite, uniting sun (male) and moon (female), holds compass and square: the compass for spirit, male, the triangle, and the square for matter, female, the four-sided figure. And what does the sphere at the bottom mean, with its triangle and square and the numerals 3 and 4? Looking closely, the numerals 1 and 2 also stand at the ends of the vertical line through the centre of the sphere. So once again 1 + 2 + 3 + 4. Evidently a geometric law is being pointed out: densely packed spheres, when their centres are joined, form a lattice of triangles and squares (see Space filling without gaps and The signatures of the cathedral builders).
The three
No number turns up in Freemasonry as often as three. There are three degrees, Entered Apprentice, Fellow Craft and Master. There are three great lights, the Volume of the Sacred Law, the square and the compasses. And three pillars support the lodge. Prichard already has:
Q. What supports a Lodge?
A. Three great Pillars.
Q. What are they called?
A. Wisdom, Strength and Beauty.
Q. Why so?
A. Wisdom to contrive, Strength to support, and Beauty to adorn.
Prichard’s pamphlet is also the first printed witness to the three degrees. Its title page names Entered Apprentice, Fellow Craft and Master, and for the first time there is a part of its own for the Master’s degree. A few years earlier the lodges had known only two degrees. The Master’s part opens with the verse “If a Master-Mason you would be, / You must rightly understand the Rule of Three”. The rule of three is also the old name for a standard method of proportion in arithmetic. So the verse can be read as a rule of calculation and as a pointer to the number at the same time.
Where the three comes from
Three traditions meet here, and they can hardly be told apart:
- Christianity. The early lodges in England were Christian, and the triangle with the eye inside it is the sign of the Trinity (see The tetractys in mysticism and religion).
- The craft. Apprentice, journeyman and master are the stages in which a trade was learned in the builders’ lodges and guilds (see The signatures of the cathedral builders).
- The Pythagoreans. Aristotle reports in On the Heavens (I 1): “As the Pythagoreans say, the whole and all things are determined by three. End, middle and beginning have the number of the whole, and this number is the triad.” That is why, he adds, this number is also used in the worship of the gods.
That the threes of the lodges come directly from the Pythagoreans is not documented. What is documented is only that early Freemasonry thought in the terms of ancient geometry, as Prichard’s four principles show. To call this Pythagorean is an interpretation.
What geometry has to say
Geometry cannot prove the symbolism. But it shows why three in particular serves so well as an image of beginning, firmness and spirit.
Three is the first surface. In the tetractys the single point is the beginning, two points determine a line, and with three points the first surface appears, the triangle. Prichard’s third principle, the superficies, is exactly this third row (see From point to solid). Fewer than three corners enclose nothing.
The triangle is rigid. Build a four-sided frame from four laths with hinged corners and it can be pushed into a rhombus. A triangle of three laths stays as it is. The reason is a theorem that Euclid proves in the Elements I 8: three side lengths fix a triangle completely, its angles included. For a quadrilateral the four sides do not fix the angles. That is why timber frames, roof trusses and bridge girders are made of triangles. For builders the triangle is, quite literally, the form of strength.
Laths of equal length, joined at the corners so they can turn. The triangle cannot change its shape, the square tips over into a rhombus.
The triangle is half a circle. The angles of every triangle add up to 180 degrees, half a full circle (proven in Angles). The quadrilateral gives a whole one. This brings us back to compass and square in the Rebis: the compass stands for three, spirit, the square for four, matter. This assignment is a later esoteric interpretation in the spirit of Hermeticism, it is not found in the ancient Hermetic writings. Added together the two give seven, multiplied twelve, and laid over each other, triangle and square are part of a twelve-sided polygon (see The signatures of the cathedral builders).
Whoever sees compass and square laid into each other in the Masonic emblem can therefore also see three and four in it, the last two rows of the tetractys. With one and two before them they make ten, 1 + 2 + 3 + 4. This is an interpretation, not evidence. But it has geometry on its side.
The four elements in Emil Stejnar
For anyone interested in the esotericism of Freemasonry, the books of the mystic and Freemason Emil Stejnar are recommended: Exerzitien für Freimaurer (Exercises for Freemasons, 1984) and Die vier Elemente. Der geheime Schlüssel zur geistigen Macht (The four elements: the secret key to spiritual power). Stejnar’s main theme is the doctrine of the four elements and its correspondence with the four-letter name of God, YHWH. What is meant are the four elements of the “microcosm”, their counterparts in the human psyche. He takes as his model a four-pole magnet, an image going back to the magician Franz Bardon (Initiation into Hermetics), whose poles depend on each other:
| Element | Effect | Quality | Power of the soul |
|---|---|---|---|
| Fire | energy | active | willing |
| Water | resonance | passive | feeling |
| Air | information | light | thinking |
| Earth | storage | heavy | awareness |
One of Stejnar’s drawings shows four pairs of interlocking hands with the name of God at the centre: the four elements are inseparable. The same motif appears in the Secret Symbols of the Rosicrucians.
Alfried Lehner
Another Freemason and author who has studied the Pythagorean brotherhood and the lambdoma is Alfried Lehner. Worth reading are his book Die Esoterik der Freimaurer (The esotericism of the Freemasons) and a second, hard-to-find work, Das wunderbare Wissen vom Wesen der Welt (The wondrous knowledge of the nature of the world), a ritual guide for Master Masons that deals among other things with the philosophical aspects of the lambdoma.
Oswald Wirth
Books on the original Christian-esoteric Freemasonry and its roots in Pythagoreanism, Platonism and Neoplatonism, which also make today’s rituals more understandable, are few. A worthwhile standard work is the trilogy by Oswald Wirth, three volumes corresponding to the three degrees of Apprentice, Fellow and Master (first German edition 2014, Edition Königliche Kunst). In Germany Wirth is known mainly as a tarot researcher and hermeticist, in France he was an important Freemason and Masonic writer. His volumes contain much on esotericism and alchemy in Freemasonry and derive its symbols geometrically, from Pythagorean and Platonic sources. He touches, if briefly, on the ten-point triangle of the tetractys and comments on the circle with a point in the centre, the ten sefirot of the Kabbalah, the four-letter name of God, triangle, square, pentagram and hexagram. He understands the cubic stone not only as a building stone but as a construction that comes about through crystallisation. What philosophy lies behind this only becomes clear through studying the cubic crystal systems.
The parts and the whole
Complex things cease to exist as soon as they are taken apart into their elements, for they are only recognised by their relations to other things. A symbol too is defined only by its difference from similar ones, and for that one must know the system in which it is embedded. This site wants to encourage readers to explore that system in nature for themselves.
On Freemasonry see also The Temple of Solomon, Herder’s Metacritique and The throne of God and the number 24.
Circle, triangle and square around one centre: the geometric framework from which the builders' lodges are said to have derived their marks. This theory of the mother figures goes back to Franz Rziha (1883) and is disputed.
Sources and further reading
Sources
- Samuel Prichard, Masonry Dissected, London 1730, scan at archive.org
- Aristotle, On the Heavens I 1, 268a
- Euclid, Elements I 8
- Heinrich Nollius, Theoria philosophiae hermeticae, Hanau 1617
Further reading
- Oswald Wirth, Le livre de l’apprenti, du compagnon, du maître, German edition Edition Königliche Kunst, 2014
- Alfried Lehner, Die Esoterik der Freimaurer (The esotericism of the Freemasons)
- Emil Stejnar, Exerzitien für Freimaurer (Exercises for Freemasons), 1984, and Die vier Elemente (The four elements)
- Franz Bardon, Initiation into Hermetics, German original Freiburg 1956