Figure 4.6
Paradoxes that led to the Polar Coordinate Model (cf. Fig. 4.5 for symbol key and context).
a. According to the GDC Model a fragment�s decision to regenerate vs. duplicate should depend only on its edges, but quadrants (cornered at the inferred peak of regenerative potency) violate this axiom -- e.g., the α quadrant duplicates, but a piece with the same vertical edge (3/4 complement to β) regenerates.

b. Dorsal (02) fragments only make dorsal structures, and ventral (68) fragments only ventral ones, but when 02 and 68 pieces are scrambled together, they regenerate middle (26) structures. (See Figure 4.5 for the numbering scheme.) This result is also hard to explain in terms of free edges.

c. The Polar Coordinate (PC) Model avoided both these paradoxes. Based on the phenomenon of intercalation (b), this model argued that growth depends on interactions between wound edges -- not on free edges. Every cell is supposed to have a circumferential (clock-face) coordinate (= angle from a 12/0 reference line) and a radial (letter) coordinate (= distance from center). The 'Shortest Intercalation Rule' asserts that when cells touch (via healing), they assess each other's coordinates and fill in missing values by the shorter route (no discontinuity at 12/0). E.g. (at right; cf. b), when 10 abuts 8, a 9 fills the gap, instead of '11, 12/0, 1 ... 7' (the longer route). Prime marks denote new values.

d. The PC Model obeys the Reciprocity Rule (see α in a). E.g., when a fragment contorts to heal 9 and 12/0 edges together, this contact creates 10 and 11 (black tissue), regardless of whether the edges belong to a quadrant (left) or its 3/4 partner (right). Thus, quadrants duplicate because none has enough angular values (i.e., > half) to regenerate. The various panels are adapted from [524-526, 1775].