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Geometry, Ratio and Freemasonry


“Philosophy [nature] is written in that great book which ever is before our eyes -- I mean the universe -- but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.” - Galileo Galilei


 Geometry is the basis of Freemasonry.  One definition of geometry offered by the Oxford English Dictionary is, “the shape and relative arrangement of the parts of something.” The word comes from the Greek: geo, ‘Earth’, and -metry, or ‘measurer’.  From this we can see that the Greeks viewed this system of mathematics as a way to measure not only the Earth, almost everything on Earth and around it. Geometry as a method of measurement not only polished Ancient Greek architecture, it also enabled the construction of massive cathedrals and fortifications in Medieval Europe and remains the fundamental mathematical system beyond arithmetic.   

 Euclid of Alexandria, born around 325 b.c.e., systematically explained the principles of geometry in the book, “Elements”.  He was a student of Plato thought to have lived and worked in the library of Alexandria. During the same period, Archimedes, native to Egypt, but living in Syracuse of modern Sicily, applied geometric principles to show fluids displace the same volume of water of any immersed object’s volume.


Geometry and Architecture


 Building on these and many other Grecian insights, the Roman Vitruvius, b. 90 b.c.e., wrote a text that survived, like the Elements, through Muslim scholars preservation during the Dark Ages.  On Architecture (De Architectura) shaped not only Roman, but Renaissance lives and later generations through its systematic approach to knowledge. If the name Vitruvius is vaguely familiar, you’ve probably seen the sketch by Da Vinci, The Vitruvian Man:  

 Vitruvius defined architecture not only as a catalog of tools, materials, labor estimates, site selection, and artistic beauty, etc, but as a system that also considers: Mathematics (geometry, ratio and proportion); Principles and divisions of architecture, town planning and temple site selection; The Doric, Ionic and Corinthian architectural orders, as well as various types of temples and altars; Water and its conveyance; Astronomy and its relevance to architecture; Construction machines, tools used to measure, and weapons of war, e.g. catapults.

 Here we begin to see that to the Ancients, architecture was much more than the commission of a cozy beach house construction by the seaside.  It formed the essential framework to both successfully sustain and defend a city, kingdom or empire by blending with nature.


The 47th Problem of Euclid

   Stepping back into the realm of the abstract, the Pythagorean theorem, or 47th principle of Euclid, has fascinated eons of scholars.  There is a bit of contention surrounding its origin, as Pythagoras lived before Euclid, and many scholars believe both the Egyptians and Babylonians had a great appreciation of the importance of the 3-4-5 triangle.  

 So, what’s the deal?  A triangle has three sides, but right triangles have special properties.  If there is one ninety degree angle and the remaining angles are not the same, each side of a triangle will have a different length.  The 3-4-5 triangle is used to illustrate how the ratio of the two sides, the height and base (or length), which can be easily measured, can reveal the hypotenuse length, or longest part, of a 3-4-5 right triangle without direct measurement.  

 Each side of a 3-4-5 triangle can be thought of as a square, subdivided into smaller squares corresponding to each length. So, if we have a side of three, imagine a box with three rows of three squares, and three columns of three squares.  Now imagine the same box, but with four instead of three squares. Three ‘squared’ is equal to nine, and four ‘squared’ is equal to sixteen. If we add nine and sixteen, the sum is twenty-five, or five ‘squared’. Taking this abstraction further, if we know two lengths of a right triangle with two non-equal remaining angles, we can determine the unknown length using math only.  This can be used many ways, from determining the length or height of an aqueduct, to plotting travel courses and finding other lengths.


 The Golden Mean


   This brings us to the last part of our brief tour of geometry. Ratio and proportion are at the top of Vitruvius’ list of critical features of architecture.  Phi (𝞥) is the 21st letter of the Greek alphabet and is used to symbolize many things, a plane angle, polar coordinate, or in this case, the Golden Ratio.  Phidias, b. 500 b.c.e., a sculptor credited with the statues of the Parthenon, studied this ratio extensively. The Egyptians are thought to have used Phi as well as Pi to construct the pyramids.  Pi  is a great story in itself, but Phi is seldom mentioned.  Plato felt it was the most important ratio and the key to understanding the proportions of the universe.

  The Golden Ratio Phi was called, “The Divine Proportion,” and was used extensively by both Ancient and Renaissance painters, sculptors and architects.  What makes it so divine? It is everywhere. Flower petals, seed heads, pine cones, tree branches, shells, spiral galaxies, hurricanes, the list is endless.  



Without going into too much mind numbing detail, the Golden Mean is the ratio between a larger and smaller side of a figure that is equal to approximately 1.618.  The Fibonacci sequence is a series of numbers based on adding the initial number to the sum of the previous set: 1 + 1 = 2; 1 + 2 = 3; 2 + 3 = 5... as the sequence grows, if you consecutively divide the larger by smaller numbers, the ratio asymptotically stabilizes at 1.618.  Mathematically defined, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Unsurprisingly, KST and the Ark both used the Golden Mean according to measurements cited in the Bible.




  From this we can deduce that our world is not only able to be described mathematically, but the use of geometry can reveal some of the most fascinating insights into art, architecture, astronomy, music, as well as solving difficult problems through abstraction.  



Geometry is truly the basis of our art and a steadfast aid in comprehending a part of the Divine,


Brandon West, S.D.


Suggested further reading:


On Architecture free e-book

Learning from Leonardo