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The Infinity Within

Nested Circles
Taiji ("The Yin-Yang")
Ad Triangulum & Ad Quadratum
Basic Heptagon
Basic Nonagon

Nested Circles

Follow the first four steps of the Basic Pentagon construction until you have a circle with two smaller circles inside of it:

Notice that the diameter of the smaller circle (shown here in red) is half the length of the larger circle (shown here in blue):

Let's make another small circle of the same size and place it within the center:

We now have a representation of "The Inverse-Square Law". What does this mean? Many things, such as light and sound, get weaker in intensity the farther away that they emanate from their source. If we double that distance, then the radiation is one-fourth the strength at that point. In the same way, while the diameter of the blue circle is twice as long as the red circle, the area of the red circle (i.e.: the amount of space that it takes up) is actually one-fourth of the blue circle.

The Pythagoreans gave this geometric pattern a spirital meaning. It represents the process by which the Universe comes into being and how we can come into balance with it. The starting circle symbolizes Unity, the Oneness from which everything finds its origin. The two smaller circles inside of it symbolize Duality, the separation that appears as the result of dividing into many seemingly "different" things. The third circle that connects them all together symbolizes Harmony:

Taiji ("The Yin-Yang")

If we repeat the above construction a few times on the same circle, then we can form the Taiji symbol of Taoism (more commonly known as the "Yin-Yang"):

Ad Triangulum & Ad Quadratum

Ancient architects built many things utilizing the proportions inherent to the triangle and square. For example, in reference to Roman temples and Medieval cathedrals, one sometimes comes across the Latin terms "Ad Triangulum" (meaning "of the triangle") and "Ad Quadratum" (meaning "of the square") to describe these specific systems of geometry.

If we look at the how triangles and squares nest within one another, we can see the doubling and halving patterns that we covered above:
The blue circle is double the size of the red circle. Inversely, the red circle is half the size of the blue circle.

This is useful to be aware of as layouts are sometimes based on these doubling and halving relationships.

The rest of the constructions on this page are approximations of different shapes with varying levels of accuracy. All of them follow the Ad Triangulum in some way.

Basic Heptagon (7-Sided Shape)

[Note: These instructions are adapated from the ones given on the Medieval Architectural Geometry website by Colin Joseph Dudley.]

The prefixes "hept-" and "sept-" mean "seven", so you may also see this referred to as a "septagon".

Step 1: This can be thought of as a simple extension of the Basic Triangle construction. Make a circle centered on either the left or right corner of the triangle. Its radius should be the distance between that corner and the vertical line.

Step 2: The circle made in the previous step crosses the central circle at two points. Use these points to continue to make circles of the same size along the circumference of the central circle.

Step 3: This makes seven points along the circumference of the central circle that we can use to form a heptagon.


There are two unicursal heptagrams {7/2} and {7/3}:

Basic Nonagon (9-Sided Shape)

[Note: These instructions are adapated from the ones given on pg. 95 of the book Quadrivium by Wooden Books.]

The prefixes "non-" and "ennea-" mean "nine", so you may also see this referred to as a "enneagon".

Step 1: This can be thought of as a simple extension of the Basic Hexagon construction. Use it to draw a {6/2} hexagram made of two triangles.

Step 2: Connect the opposite corners of the two triangles with lines.

Step 3: Use where the lines from the previous step cross the sides of the triangle to draw three circles.

Step 4: We can use the six points made by the circles in the previous step and three points from one of the triangles to create a nonagon.


There is one compound nonagram {9/3}, composed of three triangles:

There are also two unicursal nonagrams {9/2} and {9/4}: