This blog post presents a mathematically elegant and minimally arbitrary method for modeling a concrete tetrapod. The author's approach is based on constructing the shape within a cube, with the tetrapod's four legs aligned with the axes of an inscribed tetrahedron. Through an experimental process, the author discovered that using half the cube's side length as the primary parameter for both the leg's length and diameter results in proportions that align with the original patent specifications. The detailed recipe involves defining the leg's geometry using two truncated cones. These cones are derived from "scaffolding" constructs—an "axis cone" and a "face cone"—and are bounded by four key circles (waist, thigh, ankle, and sole). The author concludes that the tetrapod's form feels less like an arbitrary design and more like a shape discovered through pure geometry.
A Geometric Recipe for Drawing a Tetrapod
Core Concept: The tetrapod is constructed within a cube centered at the origin. Its four legs point from the center towards four of the cube's vertices, following the axes of an inscribed tetrahedron.
The Key Parameter - "Size": A single measurement, "size," defines the model. The author's key discovery is that if "size" is set to half the cube's side length, it naturally defines both the length and the main diameter of each leg, satisfying the 1:1 ratio described in the original patent.
Geometric Scaffolding: The construction relies on two fundamental cones to define the leg's shape and chamfer:
Axis Cone: An imaginary cone with its apex at the origin whose sides just touch the x, y, and z axes. This cone is used to locate the leg's thickest point.
Face Cone: A cone with its apex at a cube vertex whose sides touch the cube's faces. This is used to define the chamfer at the end of each leg.
Defining the Leg with Circles: The profile of a single leg is constructed using four circles along its axis:
Thigh: The widest part of the leg, with a radius of size / 2.
Ankle: A narrower point that defines the leg's taper, with a radius set to size / 3 to achieve an approximate 10° taper.
Sole: The circle at the end of the leg, defining the flat part of the foot.
Waist: An extrapolated circle at the origin, used to model the central core where the legs meet.
Final Construction: A single leg is modeled as two truncated cones (one from waist to ankle, another from ankle to sole). The other three legs are created by rotating this master leg. The legs are then clipped at the planes where they intersect at the center to create the final, solid object.
Key Takeaway: The author concludes that the tetrapod's iconic shape is not just a functional design but an elegant geometric form that emerges naturally from the relationship between a cube and a tetrahedron, making it feel "discovered rather than designed."