Three Horizons and the Making of a Stable Relation

What lets a changing pattern remain?

QuestionWhat are the three circles being used to establish first?

AnswerThey provide a chosen geometric construction in which centres, boundaries and relationships can be followed exactly. The branch first establishes that shared geometry, then asks what interaction a moving physical system would need to sustain a bound state; the circles are not yet atoms or Minds.

Explanation

Follow three circles into stability, wave interference, orbital equilibria, atomic binding and the history-bearing relations of Mind.

Supplied narration with interactive teaching models and a separate current observer clarification. The models retain their individual mathematical and physical assumptions. Audio chapter cues are approximate.

A mark appears, and then it disappears. Another returns in a recognizable way. What has changed between a passing occurrence and something whose continuation we can follow?

That is the question behind these three horizons. We are looking for a relationship that belongs to several participants together, and then asking whether such a relationship can help explain how a physical structure holds.

Begin with the circles on the screen. Each has a centre and a boundary. For this representation, we can follow what lies within each boundary, what lies outside it, and how one bounded expression relates to another.

The circles are not yet atoms or minds. They give us a simple construction in which we can see exactly what follows from the relationships we choose. First we will discover the shared geometry. Then we will ask what a moving physical system would have to do to sustain a bound state.

Two boundaries give a relation of scale

QuestionWhere does the external centre of similarity come from?

AnswerFor the unequal radii in this construction, the common external tangents meet beyond the smaller circle. The right triangles at their contact points are similar, so distances from that meeting point to the centres have the same ratio as the radii. This geometric reference alone does not show that a physical object would remain there.

Explanation

Take the first two circles. Lay a straight line against both boundaries without cutting through either circle. Then do the same on their other side.

Extend those two lines. Because the circles in this example have different radii, the lines meet beyond the smaller circle.

Watch what that meeting point means. From there, a ray touches one circle and then the other. Draw a radius to each point of contact. Both radii meet the tangent at a right angle, so the two triangles have the same angles. One is a scaled version of the other.

The distances from the meeting point to the centres therefore carry the same proportion as the radii. If the radii are two and three units, those two distances are in the ratio two to three.

This point is called a centre of similarity. It was not selected first and then used to arrange the circles. Its position follows from the scale relationship between them.

We have found a geometric reference for the pair. Whether a physical object would stay there is a later question.

P_{ij}=(r_i O_j-r_j O_i)/(r_i-r_j)

|P_{ij}-O_i|/|P_{ij}-O_j|=r_i/r_j

The third circle relates the relations

QuestionWhat remains when the three circles move or change radius?

AnswerIn the ordinary, nondegenerate construction, the three external pairwise centres of similarity remain collinear, even though the line itself can move and turn. The chained scale comparisons are consistent without making the circles equal or adding a fourth body to hold them together.

Explanation

Now construct the same reference for the second and third circles, and for the third and first.

There are three pairwise meeting points. Draw the line through the first two. The third lies on it as well.

Move a circle. Change a radius. Recalculate the three meeting points. While the construction remains in its ordinary, nondegenerate form, the three continue to be collinear. This is Monge’s theorem.

The line itself can move and turn. What remains is not one fixed position, but the alignment of the three related positions.

Why should that happen? Read the scale around the three circles. The first radius compared with the second, the second compared with the third, and the third compared with the first complete one consistent chain. Two-thirds, multiplied by three, multiplied by one-half, gives one.

The circles have not become equal. Their differences remain. What closes is the comparison around the whole.

Together with the placement of the meeting points on the extended centre-lines, that consistency gives the alignment. The line is not a fourth body added to hold the circles together. It makes a relationship among their pairwise relationships visible.

(r_1/r_2)(r_2/r_3)(r_3/r_1)=1

\det[P_{12}^{h},P_{23}^{h},P_{31}^{h}]=0

How the explanations connect

Atoms, elements and chemical relations offers another angle on What lets a changing pattern remain?. Return from the atomic binding examples to a separate three-circle construction, then ask what additional interaction a physical state would need to persist.

What lets a changing pattern remain? continues with Two boundaries give a relation of scale. After declaring the circles as a geometric construction, compare the first two boundaries through common external tangents.

Two boundaries give a relation of scale continues with The third circle relates the relations. Construct all three pairwise scale references, then examine their shared alignment.

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