Seven circles of one radius, each centred on the rim of the first, so every circle passes exactly through the centres of its six neighbours. Each circle is a phase oscillator. Two are coupled if and only if they overlap. Where they overlap, the line is lit by how closely the two agree.
We are not separate observation layers. We are united ones. The layers of observation overlap.
The construction
One rule fixes the whole figure: same radius, centre on the neighbour's rim. The boundary is drawn at twice the radius, so each circle is half its diameter. Apply the rule again at every lattice point and the figure grows 7 → 19 → 37 — not a smaller copy inside a larger one, but one rule holding at every extent. That is the self-similarity this card is named for.
This card used to show an Apollonian gasket: tangent circles, disjoint interiors, a strict hierarchy, no motion. Descartes' circle theorem is exactly as true as it ever was, and the old card is kept at /archive/self-similarity.html. What changed is what this site wants to say about how things relate.
The dynamics
dθi/dt = ωi + (K / di) Σj∼i sin(θj − θi)
The Kuramoto model on the network the geometry itself draws. j ∼ i means the two circles overlap; di is how many neighbours circle i has. The larger K, the harder they pull each other into step. The larger σ, the spread of natural rates, the harder they resist. Here K = 1.6 and σ = 0.6, and the seven hold.
r eiψ = (1/N) Σj eiθj
The order parameter r is the bar at the left: 0 when the phases are scattered, 1 when they move as one. It is an average over the parts that lives in the same space as the parts — the whole is the same kind of object as the things it is made of.
Reading the picture
The grey lines are the structure and never change. Each circle's own line brightens a little at the top of its cycle. The white is agreement: the arc one circle draws inside another lights up as the two fall into step, and goes out as they drift. When every arc is lit, the whole figure is — that is the Seed. The gold circle is the seed the rest were drawn from: the same radius, the same weight, the same rule. A peer with a name.
The card opens whole and stays whole: at this coupling the seven lock and hold. Press Scatter phases to throw them apart and watch them find each other again. Grown to 37 circles the same K would not hold — a locally coupled lattice needs more coupling the larger it gets — but that is a later card.
Live state
r = 0.00
circles = 7
overlaps = 12
K = 1.60
σ = 0.60
Lineage
The figure is old; its names are not. The seven-overlapping-circles grid is first attested
on an Assyrian palace threshold of the 7th century BCE (now in the Louvre), then on Roman mosaics
of the 1st century BCE, on the Cosmati pavements of Westminster Abbey (13th century), in
Leonardo's studies in the Codex Atlanticus, and in Alpine folk art as the
Sun of the Alps. It appears across many cultures that had no contact with one another,
which is what you would expect of a figure that falls out of a compass and one rule.
The red-ochre drawings on the columns of the Osireion at Abydos, the ones usually called the
oldest, are graffiti from the Greek and Roman centuries — not part of the temple's
decoration and not found in native Egyptian ornament. The names Seed of Life and
Flower of Life are modern, from the New Age movement, and are commonly attributed to
Drunvalo Melchizedek's The Ancient Secret of the Flower of Life (1999). The geometry here
is drawn as geometry, and nothing is claimed for it beyond what is constructed on screen.
The dynamics are the Kuramoto model (Yoshiki Kuramoto, 1975) on a hexagonal lattice with
nearest-neighbour coupling — standard, and not original to this page. That the seven-circle
Seed locks at a coupling the 37-circle figure will not is the known behaviour of locally coupled
lattices, whose phase-locking differs from the all-to-all mean field
(Hong, Park & Choi, Phys. Rev. E 72, 036217, 2005).
Assyrian threshold, 7th c. BCE (Louvre) · Roman mosaic, 1st c. BCE · Cosmati,
Westminster (13th c.) · Leonardo, Codex Atlanticus (1478–1519) ·
Kuramoto (1975) · Hong, Park & Choi, PRE 72, 036217 (2005).
About Yoshiki Kuramoto →