Part 3 · The Memory Loop and the Opening into Three Dimensions

Internal proportions

Keep the circle and square around the figure. We have used them to distinguish a field of possibility from the conditions in which an expression becomes situated. Now look within that frame. What is carried from one condition into another, so that the person who changes remains connected to the life that came before?

This is where the golden ratio returns in the proposed reading. We follow five horizontal levels across the body, from the navel through the chest toward the hairline, and compare their intervals with the nested lengths of a pentagram. The proposed comparisons include the top of the chest, the base of the breastbone, and the span of the ribcage. The question is whether one proportional relationship recurs within the figure. The golden construction is exact; its fit to every mark in the drawing remains a proposed correspondence.

Draw five equally spaced points on a small circle beside the figure. Join each point to the one two places farther around. A five-pointed star appears. Where its lines cross, they divide one another into related lengths. Choose one of those divisions. The whole line relates to its longer part in the same proportion that the longer part relates to the shorter.

That proportion is the golden ratio, phi: approximately one point six one eight. It is a relationship between lengths, not the length of one particular piece. Enlarge the construction, and the lengths change together while the ratio remains. Follow its nested golden divisions, and the same relationship appears at a smaller scale.

This is what the pentagram comparison is asking us to look for in the body: not a separate preferred number for every measurement, but a relation that can remain recognizable across different intervals. The proposed overlay opens that question. Now we need to understand why this kind of recurrence matters to the observer we are trying to describe.

Two-state inheritance

A pattern drawn at several scales is one form of recurrence. A state carrying something from an earlier state is another. To see the second, begin with two contributions, each of weight one. For the next step, add them. The result is two. Keep that result together with the previous leading contribution, one. Their next sum is three.

Repeat the same operation. Two and three give five. Three and five give eight. Five and eight give thirteen. The familiar sequence appears: one, one, two, three, five, eight, thirteen. At each step, a result that was the new whole becomes one contribution to the next whole. What came before has not simply been discarded.

Now compare the successive values rather than their total size. Three divided by two is one point five. Five divided by three is about one point six six seven. Eight divided by five is one point six. Thirteen divided by eight is one point six two five. The ratios approach phi from alternating sides.

The new whole keeps growing, while the relationship between its inherited contributions becomes more settled. Express those contributions as shares of each new whole, and they approach about sixty-one point eight percent and thirty-eight point two percent. Both remain necessary to this rule. The smaller contribution has not become irrelevant because the other carries more weight.

This is the exact golden result of the two-state additive inheritance used in the model. It does not require a leftover caused by imperfect arithmetic. Each sum is complete. What continues is a relationship through which an earlier result participates in a later one.

From inherited state to observer return

That gives us an important part of the connection to memory: the present is not required to begin without an inherited condition. But an observer needs more than a running sum. It needs the earlier condition to become distinguishable within a comparison, and that comparison to matter to what happens next.

Think again of the photograph. The face in it is available to you now, while something of the earlier life is retained within your present understanding. You notice a difference between how you understood yourself then and how you understand yourself now. That difference may change what you ask about the picture, which memory you revisit, or how you approach another encounter.

The photograph illustrates a developed observer. In the Y-Theory model, the underlying operation is made explicit: local changes contribute to a whole-level reading; an earlier reading is retained; the present and retained conditions are compared; and the retained comparison returns to alter the same field's next operation. One integrated comparison is an observation. The continuing comparison-and-return is the observer.

Reading clarification: This supplied narration uses “observer” for the comparison-and-return loop. In current Y Theory, that loop supports observation within a whole internally organized Mind: one that relates its own changing patterns and history to its outside, can intentionally change its participation, and can explore or expand awareness. The diagram illustrates supporting operations; it does not implement that whole Mind. Observation and a whole Mind.

No second person has to be placed behind the first to inspect the comparison. The present condition, the retained condition, and their active relation perform that work within the model. Nor does the observer sit inside the number phi. Fibonacci inheritance supplies a particular way of carrying contributions; whole-level comparison and consequential return supply the further operation that makes this an observer account.

The three constants now have distinguishable roles in our interpretation. Pi gives the circular reference its angular measure. The chosen e-based square keeps the manifested frame connected to the growth-and-loss model introduced earlier. Phi gives the two-state inheritance its limiting proportion. The internal pattern becomes relevant to the observer because the past must remain consequential to the next response, not because measuring a golden length would by itself reveal a mind.

Two postures, one reference

Now look down the same figure toward the two positions of the legs, and keep the two positions of the arms in view. One body has been drawn in more than one posture. We can hold the configurations against a shared reference and notice what changes. The drawing offers that comparison; in a running model, we can add an order, retain the earlier condition, and follow its consequence.

Begin with the upright stance. Its central axis meets the ground at a right angle. Leonardo's accompanying notes place the beginning of the genitals halfway up the total height. The distance from the sole to below the knee is one quarter of that height, and the distance from below the knee to the genitals is another quarter. The head, measured from below the chin to the crown, occupies one eighth.

These measurements are not all positions counted upward from the feet. The head's eighth is a length at the top of the figure. What they share is the whole body used as a reference. Mark a half, divide it into quarters, and divide again into eighths. The same operation opens finer distinctions without increasing the total height.

That is the binary reading of this part of the diagram: repeated division into two. We can label each branch zero or one and use successive choices to address a smaller interval. The labels tell us how to locate a part. They do not turn the drawn body into literal computer bits; they give our later matrix a way to preserve the route from the whole to its finer addresses.

The equilateral opening

Now open the stance. Leonardo's note describes the spread limbs in relation to the navel and says that the space between the legs forms an equilateral triangle. Follow that named shape into our geometric reconstruction. Two sides open apart, and a third joins their ends. The equal sides give three equal angles, each sixty degrees.

For the circular construction, place the apex at the navel, the centre of our circle, and put the other two vertices on its circumference below. This placement is an explicit choice in the reconstruction. It makes the two sides radii and their included angle a central angle. Keeping the point of reference visible matters: an angle at another apex would not automatically cut the same fraction from this circle.

With that centre fixed, the sixty-degree opening occupies one third of the lower half-turn, which measures one hundred and eighty degrees. Compared with the complete turn, it is one sixth. These fractions describe angular reach. The straight triangle and the curved sector fit between the same rays, but they do not enclose the same area.

Bring back the vertical centre-line. It divides the equilateral triangle into two matching halves. At the base, it meets the joining side at ninety degrees. At the apex, it divides sixty degrees into thirty and thirty. Each half is therefore a thirty-sixty-ninety right triangle.

The binary division and the triangular relation now meet in one construction. Halving opens the triangle without erasing the larger form to which both halves belong. The one-third reading of the half-turn gives a numerical correspondence with the triad. Inside, outside, and relation remain the roles through which we interpret it; they are not three equal substances produced merely by measuring an angle.

Sixfold extension

A 2025 paper by Rory Mac Sweeney in the Journal of Mathematics and the Arts develops a geometric interpretation around Leonardo's equilateral-triangle instruction. It connects a sixfold pattern with a proposed tetrahedral reading of the figure's proportions. We can examine the constructions themselves while keeping that historical interpretation distinguishable from what the geometry proves.

Take the centre-and-two-radii triangle we have just made. Turn a copy through sixty degrees and retain it beside the first. Continue until there are six triangles meeting at the same centre. Their outer edges make a regular hexagon. If we then join alternate outer vertices, two interlaced triangles appear. The sixfold arrangement is one construction; those additional star lines are another visible step.

So far, every part remains in one plane. Repetition has extended the pattern around the centre, but has not yet enclosed a three-dimensional interior. That opens the next question naturally: what changes when these triangular faces are related through depth?

A flat net closes through depth

Arrange four equal equilateral triangles as a folding net: one in the middle, with one attached along each of its three sides. Keep the middle face still and lift the other three. Their outer corners approach one point. When the matching edges meet, the four faces enclose a volume.

This is a regular tetrahedron: a triangular base and three triangular side faces, four faces altogether. Among ordinary closed polyhedra, four is the smallest number of flat faces that can enclose a volume. The construction has given us a spatial inside and outside separated by one connected surface.

Each face still belongs to the whole, but no single face encloses that interior by itself. Their arrangement does. This is the structural connection I want to retain: a relation that is incomplete as one isolated part can participate in a wider enclosure. We have constructed geometry in space, not created physical matter by folding a diagram. The question of material stability would additionally require the interactions sustaining it.

Which tetrahedral ratio?

Now surround the tetrahedron with a sphere passing through all four vertices. Measure one edge, and then measure from the centre of the sphere to a vertex. Dividing the edge by that radius gives the square root of eight-thirds, approximately one point six three three. We have a precise pair of lengths for the number; it is not the edge divided by the tetrahedron's height.

The same value appears in an ideal close-packing construction. Place three equal spheres together, then rest another in the hollow above them. Their centres form a regular tetrahedron. In a repeating hexagonal stack, two such layer separations make one vertical repeat. Comparing that repeat with the distance between neighbouring centres in a layer gives the same square root of eight-thirds. Here the number describes a specified packing geometry.

The proposed breathing margin

Return to our chosen circle-and-square frame. Its side-to-radius ratio is the square root of e, about one point six four eight seven. The tetrahedral edge-to-circumradius ratio is about one point six three three. Their relative difference is about zero point nine six percent. Both numbers are now attached to declared measurements, although those measurements belong to different constructions.

This small difference opens the proposal I have called a breathing margin. Could a living arrangement preserve an organized form while allowing a range of variation, rather than being fixed at one ideal proportion? That is the connection I want to explore through the comparison. The calculated gap is not yet a measured biological threshold separating crystal from life. To call it the breathing room of life, we still need to show how that room permits a living relation to continue.

To give the proposed margin an operating meaning, we would need to show what changes within the range, how exchange supports the structure, and how it recovers after disturbance. The requirement is familiar from the observer passage: holding a recognizable organization and remaining able to change must work together. A permitted interval in a picture becomes a continuity band only through the operation that can sustain a return within it.

Return to the figure and its reach

We can now bring the view back to the same human figure. The inner golden construction opened the question of inheritance. Inheritance led to the distinction between carrying a previous condition and using it through comparison. The two postures gave that comparison a visible reference, and the triangular opening let us follow a flat division into a spatial enclosure.

The photograph's question has remained with us: what does a life carry as its expression changes? A proportion can help us examine a pattern; a retained consequence can help explain why the next encounter is different. The person within the frame needs both organization and an exchange through which that organization can continue to develop.

Keep the circle and square visible as the figure raises its hands. Our attention has moved through the body's internal divisions and the space between its feet. Now follow its reach outward. Where do the two surrounding boundaries cross, and what becomes visible when we examine those crossings from the person's reference?

Follow this series and its connections

Begins with: Part 2 · The Circle, the Square, and the Two Centres. Keep the chosen frame, shared ground and two centres from Part 2.

Return to the question: Part 1 · The Photograph and the Question of You. Connect inherited conditions to the original question of personal continuity.

Read alongside: Observation and the observer. Distinguish the act of observation and its supporting loop from a whole internally organized observing Mind.

Explore another construction: Three Horizons and the Making of a Stable Relation. Compare geometric enclosure with the additional interactions needed for a stable relation.

Explore more deeply: Counting relations is not enough to determine a shape. Follow the distinction between mathematical arrangement, interpreted meaning and physical operation.

Where this belongs in the journey

• Read and explore Part 3 →Vitruvian continuous narration
• All available parts →Series