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
, counting only the inclined faces generated by the indentation. A closed-form expression is derived for the exposed-surface density
, where
is the apothem of the regular cell, and the triangular tessellation is shown to maximize it uniquely for every
; when
the three configurations coincide. Because each inclined face has constant slope magnitude
, the area Jacobian is constant, so clipping the indented surface with any measurable footprint
of finite positive area gives
exactly, with no boundary-error term. For the representative case
, the hexagonal density is
times the triangular one, equivalent to a
triangular advantage specific to that ratio. An independently constructed CAD model reproduces the closed-form panel areas at both reference footprints to within
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.
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