Minimal Cell Shape-

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Given some definition of complexity, what is the least complex shape of a (master) profile that meets the engineering requirements?

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An open question for mathematicians.

Subjects

• Cell Requirements
• Minimal Shape
• Catalan Volume
• Conformal Circular Version
• Additive Manufacturing Thoughts

Cell Profile Engineering Requirements-

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The cells shall connect together in an array (X, Y)

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The array shall be resistant to distortion from tension and compression forces, bending forces, shear forces, and twisting.

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Each cell profile in the array is extrudable in the Z axis and shall be substantially thin walled as most extrusions are.

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The bulk density of a bundle of extrusions shall be controlled by the profile dimensions.

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The moment of inertia of a profile can be computed given its dimensions.  This shall control how much force is required to bend the extrusion in X and Y.

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A bundle of extrusions shall hold together at outside edges.

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Individual extrusions in a bundle shall slide with each other to maintain the array dimensions.

For Engineers and Mathematicians

Catalan Volumes-

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A Catalan Surface is a surface swept out by a straight line segment of a variable vector orientation through a path.  See Wikipedia Link

A Catalan Volume  could be defined as an array of Cells that forms a plane of which its normal follows a path with a variable direction and rotation that sweeps out a volume.

Additive Manufacturing-

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This technology may or may not be appropriate for additive manufacturing but here are some thoughts:

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Standard 3D Printers have an assembly speed on the order of n^3 (with tiny pixels).

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An extrusion array would build a volume on the order of n^2 (with cell profiles).

An X,Y array of profiles

Extrusion profiles can be designed that produce a circular(pipe) assembly.

As the profiles get smaller toward the center, their angles remain the same,

conformally.