Prismatoid Band-Unfolding Revisited

Abstract

It remains unknown if every prismatoid has a nonoverlapping edge-unfolding, a special case of the long-unsolved "Dürer's problem." Recently nested prismatoids have been settled [Rad24] by mixing (in some sense) the two natural unfoldings, petal-unfolding and band-unfolding. Band-unfolding fails due to a specific counterexample [O'R13b]. The main contribution of this paper is a characterization when a band-unfolding of a nested prismatoid does in fact result in a nonoverlapping unfolding. In particular, we show that the mentioned counterexample is in a sense the only possible counterexample. Although this result does not expand the class of shapes known to have an edge-unfolding, its proof expands our understanding in several ways, developing tools that may help resolve the non-nested case.

Figures (16)

• Figure 1: Hexagonal counterexample. Fig. 1 in o-ufncp-13.
• Figure 2: $n_B,n_A = 14,16$. Here $z = 0.2$ when the diameter of $B$ is $1$.
• Figure 3: (a) Convex angle ${\theta}$ at $b$ in the plane, angles ${\phi}_a$ and ${\phi}_c$ above the plane. (b) ${\theta}$ is the length of the blue arc on ${\mathbb{S}}$, shorter than the sum of the ${\phi}_a$ and ${\phi}_c$ arcs (brown).
• Figure 4: (a) $v$ reflected to $v'$. (b) $\vec{c}$ to $\vec{v'}$ (blue) to $\vec{c'}$ (Purple) is half a great circle, of length $\pi$.
• Figure 5: ${\phi}(z)$ when $({\theta},x,y) = (120^\circ,0,1)$.

Theorems & Definitions (9)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Proposition 9