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Geometric Shapes

The idea behind the form

The three mother figures of geometry

The three archetypes shapes


the mental triangle - passive
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the emotional circle - passive
  • Geometry of circle
  • The 22 regular Polygons of circle
  • The Flower of Life or Rosace
  • Yin Yang
  • Ourobouros
  • Symbol of Self
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the material square - passive
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« It is from the three fundamental bases: the circle (the round form), the triangle (the triangulation) and the square (the quadrature) that the most elaborate Gothic buildings are born, because without the help of the straight line, the angle, the circle, the arc, the triangle, the square and the polygon, it is impossible to build concretely. »

Franz Rziha ~ Study on the marks of stonemasons.
The 3 tools of geometry

The 3 symmetry axes

Or the different rays of spirit


triangle's symmetry axes

The triangle ~ 120° angle
Three Rays


hexagon's symmetry axes

The hexagon ~ 60° angle
Six Rays


square's symmetry axes

The square ~ 90° angle
Four Rays

Only the regular polygons whose angle at the vertex is a submultiple of 360 can satisfy the conditions of a regular tiling; we obtain the angles of 120°, 90° and 60° corresponding precisely to the Hexagon, to the Square is to the Triangle and equilateral, and there are no other solutions.

« Everything that exists in the bodily nature, all forms, the slightest features, are and can only be meetings, combinations, or divisions of the primitive signs which are numbers. Nothing can appear among the sensible things that are not written by them, that do not descend from them and that do not belong to them, just as all possible figures of Geometry will always be composed of points, lines, circles or triangles. »

L.C. de Saint-Martin ~ Le Tableau Naturel

Among the regular polygons, the triangle and equilateral, the square, the Pentagon, the Hexagon, the Octagon, and (more rarely) the Decagon, are (with the circumference), the essential elements of any architectural layout, and also play an important role in Decorative Arts.

The pentagram (Starry Pentagon) and the hexagram (Formed by two equilateral triangles superimposed head to tail) have provided symbols that are indestructible to mystical schools, to magical brotherhoods of all races.

The 3 tessellations or tilings 

Paving the plan


triangular tiling

Triangular pattern paving
Obtained by juxtaposition of hexagons


hexagonal tiling

Hexagonal pattern paving
Obtained by juxtaposition of triangles



square tiling

Squared pattern paving
Obtained by juxtaposition of squares


The three ways to cover the plan

A regular tessellation is a pattern made by repeating a regular polygon so as to cover a surface without leaving any empty space. There are only three regular tessellations, or ways to fill the Euclidean plane with regular polygons: the triangle, the square and the hexagon.

the 9 regular tessellations

The pentagon does not allow the plan to be completely fulfilled. It is not to be considered as a "mother figure". On the other hand, its axis of symmetry located at the summit does not give it an isometric character. Its study is based on a polygonal or symbolic approach.

Matyla C. Ghyka, in the foreword to her work on The Golden Number, speaks of symmetry and proportion in Mediterranean art:

« The symmetry and proportion of Mediterranean art are described as "graphic processes which (...) result in the reproduction of lines in which the theme of the whole is reflected and reproduced, according to a certain more or less veiled rhythm, in each of the parts. Law of analogy, of repetition of the fundamental form, of identity in variety, of the Same and the Similar. » »

Circle and hexagon

The side of the hexagon being equal to the radius, the circumference of the circle will be 3 cm, which is very close to the exact value of the circumference C = 2Πr, i.e. 3.14.

This is why we consider the hexagon as a "circular number".

The Hexagon can indeed be broken down into six equal equilateral triangles. It has the unique property among regular polygons to have its side equal to the radius of the circumscribed circle, which makes it very simple to construct the regular Hexagon inscribed within a given circumference. This property confers to the network of points constituted in a plane by the vertices and centers of a system of juxtaposed equal hexagons, the particularity of being perfectly "isotropic": the distances from any one of the points of the network to the six points surrounding it are equal to each other and are also inclined to neighboring directions.

To inscribe an equilateral triangle in a given circle, start by marking the vertices of the hexagon with a compass aperture equal to the radius and join three of these vertices A, B, C by skipping the intermediaries.

If one also joins the three remaining vertices one obtains the pseudo-hexagon-star or hexagram, formed of the two equilateral triangles ABC, A'B'C'. This hexagram (Seal of Solomon) plays an important role in Jewish mysticism and decorative art.


The three grids of the cathedral builders

Sacred universal patterns


triangulation grid

The spirit
The Intellectual Sphere

Pythagorean symbol of Wisdom, the triangle is also the Christian symbol of the divinity and in particular of the Trinity through the three Persons: The Father, the Son and the Holy Spirit.
The triangle also evokes the universal ternary law: time (past, present and future), three-dimensional space, the 3 kingdoms (mineral, vegetable and animal), etc. It is associated with the qualities of the spirit (thesis, antithesis and synthesis).
Symbol of the "mental body".


rosace grid

The soul
The Emotional Sphere

Symbol of the All, of eternity, the circle also represents nothingness as long as it is not animated by a central point. Symbol of the "astral body", it evokes the soul, as well as the "emotional body" or the "subtle body". Everything is a question of proportion.
In Renaissance poetry in general, the circle was a symbol of perfection and is a symbol of the human soul.


quadrature grid

The body
The Vital Sphere

The square is a symbol of the manifested world. It evokes the 4 elements (Fire, Air, Water and Earth), the 4 states of matter (igneous, gaseous, liquid and solid) or the 4 cardinal points (North, South, East and West).
It is thus an image of space and matter, of concretization and stability.
Symbol of the "physical body".

By applying the three basic movements of geometry (duplication, rotation and homothety) to the three mother figures, we obtain the three construction grids from which any number can be plotted.

It is from these three grids that we have drawn all the symbols of Geonumerology. When a symbol can be drawn on the rosette and the triangulation, their addition represents the trilobe (double) and generates symbols where intellect and emotion are mixed, and when it can be drawn on the rosette and the quadrature, their addition forms the quadrilobe which generates symbols tinged with emotion and materiality.

« As for the master of our time, he could not conceive any rose window, any pillar, any Gothic detail without an original figure inspired, certainly, by his creative imagination but built according to the same principles of repetition, shrinking, division and tilting of simple basic figures, such as the mother figures. »

Studies on the Brands of Stonemasons ~ Franz Rzihaa

The Aikido principles

Triangle, circle and square
triangle, cercle and square in aikido
The representation of Aikido principles by the geometrical shapes triangle, circle and square are taken from the Shinto cosmology theory of “Gogyo Gogen” where the elements of existence, Liquid, Solid and Gas are all shown to be transmutable and interrelated.
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