05 · Deformation retraction
From G₃ to F₃—and back.
Straighten the ε-labels, retract the negative coordinates, and watch the Goodwillie polytope settle onto the subdivided cube called the freehedron.
01
pε : G₃ε → G₃⁰
Straighten the labelsThe ε and 1−ε coordinates move to 0 and 1. Negative coordinates stay fixed.G₃ · 12 facetsF₃ · 8 facets
Mathematical reading
A deformation retraction, shown in two honest stages.
The first half is the map pε: it is determined by the labeled vertices and straightens ε and 1−ε. It should not be read as one global ambient affine projection. The second half applies the continuous coordinate clamp(x₁, x₂, x₃) ↦ (x₁, max(x₂,0), max(x₃,0)).
At the endpoint, the face x₂=0 is cut at x₁=1/3 and the face x₃=0 at x₁=2/3. The resulting cubical cell complex has f-vector(1, 12, 18, 8, 1),with four quadrilateral and four pentagonal facets.