# Overlapping Volcano Unfolding of a Prismoid

A prismoid is defined to be a skewed, truncated pyramid over a convex polygon. A volcano unfolding of a prismoid is made by cutting all side edges and all but one top edge and then flattening out the figure in the plane. This Demonstration shows volcano unfoldings of prismoids with a regular base.
In any volcano unfolding of a prismoid, no pair of side flaps overlap. So any volcano unfolding of a pyramid is free of overlaps.
But, this Demonstration shows that a prismoid has a volcano unfolding in which overlaps occur.
Even so, a prismoid always has a volcano unfolding in which no overlap occurs [1, pp. 321–323].

### DETAILS

A prismatoid is defined to be the convex hull of two convex polygons in parallel planes. Therefore, a prismoid is a prismatoid where its two defining convex polygons are equiangular and oriented so that corresponding edges are parallel.
Whether a prismatoid can always be unfolded without overlapping edges is still an open problem, as is the case for general convex polytopes.
An example of a volcano unfolding of a prismoid that overlaps is given in [1, p. 323].
An example of an overlapping volcano unfolding of a nearly flat prismatoid with eight triangular faces is shown in [1, p. 324].
An example of an overlapping volcano unfolding of a nearly flat prismatoid with five faces is shown in [2].
References
[1] E. D. Demaine and J. O'Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, New York: Cambridge University Press, 2007, pp. 321–324.
[2] I. Hafner. "Four Polyhedra, Each with an Unfolding that Overlaps Itself" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/FourPolyhedraEachWithAnUnfoldingThatOverlapsItself.

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