Scale-independence

The formation of the same structure at many different scales is common throughout biology. Internally complex organs, such as lungs and
kidneys, exist in species of widely different sizes. A fragment of hydra one hundredth the volume can give rise to an almost complete
animal, and sea urchin embryo develop normally over an eightfold size difference.

The origami shape language is ``scale-independent''; the local cell program can create the same shape at many different scales with no
modification, even though the number of cells involved may vary significantly.

The figures demonstrate this idea using the cup program. The figures (a) and (b) show the final crease patterns from generating a cup on 4000 processors and 8000 processors respectively, with no modification to the program. Figure (c) shows a four cup tesselation on 8000 processors (same sheet as (b)) which is generated by first folding the sheet into one quarter size (i.e. surface of approx 2000 processors) and then running the same cup sequence. ++

(a) (b) (c)

Scale-independence is achieved by using relative comparisons of gradients. This idea of ``balancing'' gradients was proposed by Wolpert and he used it to explain scale-independence in the initial patterning of the hydra and sea urchin. He used a small number of gradients to create a coordinate system and different comparisons (e.g. $a=2b, a=b/c$) to generate patterns. He also introduced the canonical test for scale-independence - the french flag problem, where a given area must be divided into three equal regions: red white blue, irrespective of the number of cells, t and inspite of failures (even regional failures).

Our local cell language uses the idea of balancing gradients in a different way. There are only two simple comparators: a greater-than ($>$) comparator on a single gradient used by axioms 1 and 4 and an equality ($=$) comparator on two gradients used by axioms 2 and 3. The language uses many gradients to create nested structures, by using an insight from origami. Origami is itself scale-independent. Each fold is relative to the current boundary conditions and each fold generates new boundaries, thus incrementally creating complexity while still remaining relative to the original boundary.

Our language provides a general framework for creating scale-independent patterns and shapes. It gives us an insight into how internally complex structures like lungs and kidneys could appear at a wide variety of scales (compare high school frog disection and human anatomy), without any modification to the DNA program, simply by varying the boundery conditions to reflect size.