Multidirectional shape and proportionate growth

Characterize multidirectional growth of an active tissue, including the evolution of its shape and aspect ratio, and determine whether the tissue remains similar to its initial shape during growth.

Background

The paper analyzes one-dimensional elongation and isotropic dilation, but does not treat simultaneous growth in multiple directions. Allowing shape changes introduces a family of traction-free steady shapes and permits different growth rates along different directions.

In this setting, biological proportionate growth must be reformulated as preservation of tissue shape rather than preservation of a single length ratio. The unresolved issue is whether an organ undergoing multidirectional growth remains geometrically similar to itself.

References

Growth in more than one direction at once is a harder problem than the isotropic dilation worked out in , and one we have not attempted. Once the tissue may change shape the steady state is no longer a single number, many shapes satisfy the traction-free condition, and the rates along different directions need not agree, so that the aspect ratio itself evolves. Proportionate growth then becomes a statement about shape -- whether an organ remains similar to itself -- rather than about a ratio of lengths, and we regard this as the natural next question.

— Growth phases of an active tissue: determinate, indeterminate, and proportionate  (2608.20091 - Watwani et al., 20 Aug 2026) in Discussion