One Replacement Rule Builds a Packing

Begin with one tangent quadruple

Begin with the unit enclosure and three inner circles. Their signed curvatures are minus one, two, two, and three. Every pair in this root configuration is tangent. Instead of trying to draw an entire intricate packing at once, we will repeat one local operation on sets of four circles.

Hold the three inner circles fixed and replace the enclosure in this quadruple. The new circle has curvature fifteen and radius one fifteenth. It occupies the gap between those three inner circles. The enclosure remains part of the accumulated drawing, even though it has been replaced in this particular working quadruple.

This distinction is essential: a replacement changes the four-member configuration used for the next calculation. Building a packing retains circles already found. We do not erase the enclosing boundary or every earlier circle whenever a new local configuration is formed.

The first generation also includes replacements of the other root members. Those choices fill other gaps or recover the opposite symmetric solution. Each new circle comes from three fixed tangent parents and one replaced member. The same rule is applied at different locations.

Make the local calculation explicit

Focus again on curvature fifteen. Its three parents have curvatures two, two, and three. Twice their sum is fourteen. The replaced enclosure has curvature minus one, so subtracting it gives fifteen. The radius is then the reciprocal, one fifteenth.

Keep that new circle, the left half-radius circle, and the lower third-radius circle. Their curvatures are fifteen, two, and three. Replace the other half-radius member of the working quadruple. Twice twenty minus two gives thirty-eight.

The selected circle has curvature thirty-eight, radius one thirty-eighth, and center at minus three thirty-eighths, minus twelve thirty-eighths. Its three gold parent boundaries meet it at separate contacts. The earlier circle of curvature fifteen is now one of those determining neighbors.

Pull back to see the parent and child together. Recursion means that an output from one step can become an input to the next. The rule has not changed, but the local set of values has. That reuse produces increasingly detailed configurations.

Different choices create different descendants

Return to the working quadruple with bends fifteen, two, two, and three. We previously replaced a bend-two member. This time keep fifteen and both twos, then replace the bend-three member. Predict the new curvature from these three fixed neighbors.

Twice fifteen plus two plus two is thirty-eight. Subtract the old bend three to get thirty-five. The new circle has radius one thirty-fifth and center at zero, minus six thirty-fifths. It is a sibling choice in the recursive construction, not the same circle as curvature thirty-eight.

Compare the neighboring small circles. The thirty-five and thirty-eight labels correspond to slightly different radii. Both were obtained by one more replacement from a configuration containing fifteen. The shared rule does not force all circles in the same generation to have equal size.

Return to the wider second-generation pattern. Branching here means choosing which member of a tangent configuration to replace. Several choices can lead to circles already encountered, so the native builder recognizes duplicates instead of adding another copy of the same circle.

Read generation as construction history

Show the first generation again. A generation describes the recorded replacement depth from the seed along a construction route. It is not the number printed inside a circle: that label is its curvature. The small fifteen circle and the larger six circles can share a generation.

Show the second generation. Circles now use configurations formed during earlier replacements. We can identify a chain through a specific parent, as we did from fifteen to thirty-eight. That chain is more informative than merely saying that the picture became more complicated.

Add the third displayed generation. The World retains older circles and exposes more descendants. Some very small mathematical circles are omitted by the renderer’s finite detail threshold. This image is a finite sample of the recursive construction, not every circle from an infinite packing.

Return to generation one. Removing later generations from view does not change the radii of the circles that remain. A generation filter controls which constructed objects are displayed. It differs from scaling circles, moving their centers, or changing the tangent seed.

A replacement can be reversed

Focus on the curvature-thirty-eight circle and its fixed parents fifteen, two, and three. Their sum is twenty. Twice twenty minus thirty-eight is two, recovering the replaced half-radius circle’s curvature. The same replacement operation takes us back when applied to the same three parents.

This reversibility explains one source of duplicate work. Immediately replacing the just-replaced member returns to the earlier configuration. The builder can avoid following that step again. Avoiding a repeated configuration is a computational choice; it does not alter the mathematical relation.

Reverse the thirty-five example. Its parents have curvatures fifteen, two, and two, whose sum is nineteen. Twice nineteen minus thirty-five is three. We recover the lower third-radius circle, demonstrating the same reversal with a different parent set.

Reverse the first example as well. Twice two plus two plus three, minus fifteen, is minus one. The enclosing solution returns. At each step, the unchanged parents identify which inverse operation we mean. A curvature value alone does not specify that parent set.

An image has a finite resolution

Show three generations in the whole enclosure. Every displayed circle has a positive radius and a finite curvature magnitude. An infinite packing would continue filling eligible gaps without a final smallest circle. The current display necessarily stops at finite computational and visual limits.

Approach the region containing curvature fifteen and its descendants. A camera approach changes how large these fixed circles appear on screen. It does not apply the replacement rule. More visible detail and a new mathematical generation are different kinds of change.

Return to the whole view. Counting only visible pixels would merge or miss small circles and would not provide an exact count of the mathematical packing. We therefore used explicit selected objects and parent relations for our worked calculations, rather than estimating quantities from visual density.

The picture suggests repeated structure, but this lesson does not measure a fractal dimension. Such a claim would require a specified mathematical set, definition, and method. What we have demonstrated directly is a recursive tangent-circle construction using the same replacement relation.

Reverse one replacement and distinguish the accumulated drawing

Focus on the circle with bend fifteen between the two bend-two circles and the lower bend-three circle. Its radius is one fifteenth. In this local quadruple, fifteen occupies the slot once held by the negative-one enclosure. The large enclosure still belongs to the accumulated packing surrounding this close view.

Apply the same replacement arithmetic to that slot again. Twice the fixed-neighbor sum is fourteen, and fourteen minus fifteen is negative one. The calculation recovers the enclosure. Replacement is reversible on a fixed quadruple; it is not an instruction to create a brand-new smaller circle every time we repeat it on the same slot.

Return to the enclosure and its first replacements. A recursive construction must choose another member or another local quadruple to continue into a new gap. Otherwise it simply reverses the previous step. The native packing keeps already found circle identities instead of treating a recovered circle as an additional distinct object.

Now examine the bend-thirty-five circle at center zero, negative six thirty-fifths. It comes from replacing the bend-three member while holding the two bend-two circles and bend fifteen fixed. Twice two plus two plus fifteen, minus three, equals thirty-five. We changed the slot being replaced, so the result is a different circle.

Return to the broader neighborhood. The bend-fifteen and bend-thirty-five circles have radii one fifteenth and one thirty-fifth, but generation labels describe replacement steps rather than a globally sorted list of radii. Read the parent configuration and selected slot when tracing the construction, then use the radius to describe the resulting size.

Trace a route through the packing

Return to the root as a starting point for your own trace. Keep a record of the three fixed curvatures, the replaced curvature, and the new result. That record distinguishes a justified construction from a guess based on the size of an empty-looking region.

Follow the first route to fifteen. The fixed values two, two, and three sum to seven, and the old value is minus one. The new value is fifteen. Keep its center and contacts in view so the arithmetic remains attached to a particular native circle.

Continue to thirty-eight using fixed values fifteen, two, and three and old value two. The sum twenty doubles to forty, and subtracting two gives thirty-eight. A complete route records which circle was replaced, not merely a list of increasing numbers.

Finish with the accumulated packing. One local relation, applied repeatedly to tangent quadruples, builds this network of circles. Parents stay fixed during each replacement; earlier circles remain available; duplicates are recognized; and every rendered view is a finite portion of the continuing construction.