Part 2 · The Circle, the Square, and the Two Centres

Carry the human question into the drawing

Here is that figure: one human body, with its arms and legs shown in two positions, reaching into a circle and a square.

The photograph asked what connects the different conditions of a life. This drawing opens another way to examine a person: through the relationships between the body, its proportions, and the boundaries around it. Can the inside–outside relation become something we can construct, measure, and understand—not only something we describe in words?

Leonardo drew the Vitruvian Man around 1490. It is a study of human proportion in dialogue with the architectural writings of Vitruvius. But rather than stop at recognizing the familiar image, follow what happens when we begin reconstructing it.

When I started drawing the figure myself and placing the square against the circle, one detail became a question: why do the two shapes not share the same centre?

Coincident centres then a changed relation

For a moment, set the body aside without losing its place. Draw a circle. Mark its centre. Then place a square around that same point. The shapes remain different: the circle's boundary curves smoothly, while the square's four sides meet at corners. Yet their centres now give us only one position.

Now lower the square while leaving the circle where it is. A second position becomes distinguishable. There is a separation we can measure and a direction from one centre toward the other. The overlap changes with that separation. We can follow what happens to one boundary relative to the other.

This is the difference I want to examine. The offset gives us a measurable separation; movement would require something to change that separation or another part of the arrangement. If the relation departs and returns, we can follow its oscillation against a reference. The drawing lets us distinguish the positions through which such a movement could pass.

Restore the body and locate the centres

Keep that distinction in view as the human figure returns. The circle's centre is at the navel. The square's centre is lower, around the groin. One body belongs to both constructions, but the two constructions do not read the body from the same centre.

These bodily locations also carry meaning in my reading. The navel recalls the connection through which life was received before birth. The lower centre draws attention toward reproduction and the possibility of life being carried onward. Origin and generation become two ways of reading the same figure. That symbolism is part of the interpretation I am developing through the geometry.

Place both on common ground

Now bring the two shapes onto a shared ground. In our reconstruction, the bottom of the circle and the lower edge of the square meet the same horizontal line beneath the feet. We are giving them a common reference without forcing their centres to coincide.

The reason for the offset becomes visible. From its lowest point, the centre of the circle stands one radius above the ground. The centre of the square stands halfway up its side. If that side is shorter than the circle's diameter, the square's centre must sit lower. The difference follows from their proportions once we hold the shared ground fixed.

So there are two questions within the one image. Where are the shapes placed relative to each other? And how large is one in relation to the other? The first brings us to the two centres. The second brings us to the numbers.

Read the reported comparison

The golden ratio, about one point six one eight, has often been proposed as a key to the drawing. But we need to ask which two lengths are being compared. A number can describe one relationship without describing every relationship in the picture.

Measurements discussed in research on the drawing give a circle radius of about one hundred and ten millimetres and a square side of about one hundred and eighty-one point five millimetres. Those are approximate measurements of a hand-drawn construction. Dividing the square's side by the circle's radius gives one point six five.

So the side-to-radius comparison gives us a different proportion from the golden ratio. Keep that particular relationship in view as we change the scale of measurement.

Normalize, then compare areas

Keep the drawing's proportions, but change the measuring unit. Call the circle's radius one. We have not enlarged the circle or shortened the square; we are measuring every length against the same radius. The square's side now reads one point six five in those units.

Measure around the circle and straight across its centre. The ratio of circumference to diameter is pi. When the radius is one, the enclosed area is also numerically pi—about three point one four one six square units.

Now measure the square's area using that same unit. Multiply its side by itself: one point six five times one point six five gives two point seven two two five square units.

Notice a nearby value: Euler's number, e, approximately two point seven one eight two eight. The square's calculated area is close to it, with a relative difference of about zero point one six percent. This is the numerical comparison that opens the pi-and-e reading. It is a close correspondence, not an exact equality in the reported measurements.

Select the exact pi/e reconstruction

To explore that reading, make one explicit choice in our mathematical reconstruction. Let the square's side be the square root of e—about one point six four eight seven—rather than one point six five. Its area is then exactly e, while the unit circle's area remains exactly pi. The ratio of their areas is pi divided by e.

The reported drawing gives us an approximate comparison. Our reconstruction gives us an exact rule to follow. I am using the near agreement as the starting point for a Y-Theory model, while leaving the historical question of Leonardo's intended rule open.

Return to the shared ground. With this chosen side length, the square's centre stands about zero point eight two four four units above it; the circle's centre stands one unit above it. Their separation is about zero point one seven five six of the radius. The offset is no longer merely a visual impression. It is a calculable part of this chosen relationship.

Read the circle as a possibility-facing reference

But why connect these shapes with these two numbers? What is the relationship supposed to explain?

Follow a point around the circle. It can pass through every angular direction and return without encountering a corner. Pi gives us the measure of that circular passage. The shape makes continuity of direction visible, and gives a changing phase a complete turn against which to be read.

In the Y-Theory interpretation of this image, the circle represents Consciousness: the possibility-facing openness available to an identity. It is not only the one thought or action occurring now. It points toward the compatible ways that the identity can participate beyond its present expression.

The figure's different postures help us approach that distinction. One posture does not exhaust the body's possible reach. Likewise, one present understanding does not exhaust every understanding the Mind could become capable of carrying. The circle gives us a visible reference for that openness. Source remains wider than the possibilities accessible through any one identity.

Read e and situated manifestation

Now turn back to the square and to e. This number appears naturally when a quantity changes at a rate proportional to the amount already present. With a constant positive proportional rate, the amount grows exponentially; with a negative one, it decays. What is present participates in determining how much change comes next.

The simple persistence equation in our mathematical source has that form: a represented quantity changes according to the difference between reinforcement and leakage. When those rates are constant, its evolution is expressed through an exponential. That gives e a specific connection to growth and loss under the conditions of this model.

In this reading, the square represents Manifested Structure: the definite, measurable conditions through which a possibility becomes an actual expression. Its edges give us directions against which to locate something. We can divide the frame into positions and keep track of a change from one position to another. If we add a grid, its cells give those distinctions separate addresses. That is a chosen way of representing discrete positions within the frame.

The two shapes now carry different questions. The circle asks about the wider range of compatible participation. The square asks where and how a particular expression becomes situated. Their relationship matters because a possibility becoming actual does not exhaust everything else that remains possible.

Retain symbolic and physical comparison angles

My account also uses the symbolic language of feminine and masculine for these complementary aspects. Here, those names refer to the possibility-bearing and form-giving roles in the interpretation. Both belong within the reading of one person: receiving possibility and giving expression remain related rather than becoming two disconnected worlds.

The same inquiry also opens a physical question. Could a field description and a propagating disturbance give these two roles a specified physical expression? My proposed comparison with scalar potentials and transverse waves belongs here. To develop it, we would have to identify the field, its allowed modes, and the equation connecting them. The geometric reading gives us a question to investigate; the physical relationship still has to be supplied.

Return to the person and open the next detail

For now, return to the human being between the two outlines. In this interpretation, a life is situated in definite conditions while remaining open to more than its present state. We do not experience all our possibilities at once. We meet some of them through what our body, our history, and our relationships make accessible.

That returns us to the person in the photograph. The life you recognize was never exhausted by a single expression. Something became actual, left a consequence, and changed what the person could meet next. The circle-and-square reading gives that distinction between available possibility and situated expression a visible form.

But we have so far been examining the frame around the figure. The person within it has to do more than occupy a position. How does an earlier experience remain available inside a later one? What allows a difference to become a comparison, and a comparison to change the next response?

Keep the circle, the square, and their two centres visible. Now let your attention move inward, toward the proportions marked across the body. The golden ratio returns as a new question—not as the assumed answer to every measurement, but through internal proportion and the possibility of inheritance.

What is the person carrying within the frame that allows the next moment to belong to the same life?

Follow this series and its connections

Begins with: The Photograph and the Question of You. Begin with the human question that gives this geometric inquiry its direction.

Another angle: Counting relations is not enough to determine a shape. Distinguish a geometric arrangement from a physical law governing change.

Another angle: Three Horizons and the Making of a Stable Relation. Explore another geometric construction, then examine the additional laws needed for stability.

Read alongside: Observation and the observer. A position or geometric centre alone is not a whole observing Mind.

Continue with: The Memory Loop and the Opening into Three Dimensions. Carry the same frame into golden inheritance, binary division and a three-dimensional enclosure.

Where this belongs in the journey

• Listen and explore Part 2 · 11 minutes →Vitruvian continuous narration
• All available parts →Series