Platonic Esoteric Geometry

Let none but geometers enter here

While Plato was publishing his dialogues, his Academy was researching and teaching sacred arithmetic and geometry. This explains the constant references to mathematics in the dialogues.

At the Platonic Academy of Melbourne we believe that a full understanding of Plato’s philosophy is only achieved by seeing and experiencing just how it was informed by the mystical mathematics practiced at Plato’s original Academy. This is why this very special course mixes discussion of the philosophical teaching with hands-on geometric drawing.

Join us for six small-group weekly sessions consisting of a talk, a discussion and then some drawing with compass and ruler.

For geometry beginners. Beginners with Plato might also consider taking Reading Plato.


Hands-on in Hawthorn
Saturday & Sunday 19-20 February 2022.


Instructors: Bernie Lewin and Nabeel Khan

Take this course to:

  • Practice the art that informs Plato’s dialogues
  • Get in touch with the universal form of experience through geometric drawing
  • Open your eyes to similarity and self-similarity in nature, art and architecture
  • Learn the ancient art of geometry without the tyranny of scale and without the distraction of numbers or algebra
  • Discover why geometry is the gateway to Platonism.

Course Outline

Session 1 | Let none but geometers enter here

We begin by paying attention to the geometry of experienced objects, to sense and imagine their space, shape and form, their similarity and self-similarity, and these beyond the prison bars of scale. The geometric imagination is thereby awakened so that the mind’s eye may see, as the ancient Academics once did, the very elementary form of knowledge that underlies its every communication through signs, language and reason. We then take up the compass and draw our first circles. We rule our first square.

Session 2 | The doubling of square (Meno p. 81 to 85)

Starting with the problem of the doubling of the square, we explore the marvelous symmetries of the square’s diagonal, otherwise known as the square root of two (√2).

Session 3 | Geometric expansion of the doubling square

The square that is doubled is doubled again to give our first two geometric series, by double and by √2, and these serve to demonstrate the ‘power of the root’ to generate by ratio (logos).

Session 4 | The Spiral of Theodorus (Theaetetus p. 147d)

Diagonals underlie so much of the mathematical imagination! The use of arithmetic to measure diagonals gives what modern mathematics imagines to be numbers, imperfect, unresolved and irrational, thereby confusing geometric measurement with pure arithmetic. We eskew this modern confusion to instead explore diagonals, just as the ancients did, according to their own simple, self-measuring form.

Session 5 | The Golden Ratio (Republic p. 509d)

The essence of geometry is ratio (logos), and ratio of ratios is proportion (ana-logia). There are proportions in 4-terms, continuous proportions in 3-terms, and then there is that special cut of the line in 2. We make that cut with compass and rule, and then explore the symmetries of this marvelous ratio that the ancients kept secret, that the Renaissance called Divine and that we now call Golden. 

Week 6 | Figurate numbers and musical ratios (Timaeus 35b-36c)

Arithmetic arises from its geometric origin in the one-and-its-other, alternating odd & even, other & same, through linear time. The line is the origin of the plane, and the first plane numbers emanate triangular. The plane is the origin of the solid, and the first solid numbers emanates a crystalline pyramid with triangular base. Physical counters are used to build plane and solid (‘figurate’) number series. The fourth triangular number gives the sacred Tetractys, which contains the three musical ratios elementary to the tonal scale that remains in use today. The course finishes with the Renaissance revival of Platonic Esoteric Geometry, including the architectural application of these musical ratios.