Exposed-Surface Maximization for Bioreceptive Architectural Components: Triangular Optimality in Regular Tessellated Panels

Bioreceptive building envelopes offer a potential pathway for integrating biological colonization into architectural surfaces. This article proposes a geometric design framework for bioreceptive façade components in which the surface itself is shaped to increase the exposed area available to the environment, and asks which admissible regular Euclidean tessellation maximizes that area. Triangular, square, and hexagonal tessellations are compared under a common circumscribed radius R and a common absolute indentation depth hj, counting only the inclined faces generated by the indentation. A closed-form expression is derived for the exposed-surface density σn=hj2+rn2/rn, where rn is the apothem of the regular cell, and the triangular tessellation is shown to maximize it uniquely for every hj>0; when hj=0 the three configurations coincide. Because each inclined face has constant slope magnitude hj/rn, the area Jacobian is constant, so clipping the indented surface with any measurable footprint Ω of finite positive area gives Sn(Ω)=σn|Ω| exactly, with no boundary-error term. For the representative case hj/R=45/34, the hexagonal density is 0.6454 times the triangular one, equivalent to a 54.93% triangular advantage specific to that ratio. An independently constructed CAD model reproduces the closed-form panel areas at both reference footprints to within 10−3% and preserves the same ordering. Exposed area is a geometric descriptor of the available surface–environment interface and a candidate design parameter, not evidence of colonization, water behavior, or durability; no material, fabrication, or biological validation is claimed.