From flat sheet to cone without cutting: the wrinkled cone*

It is relatively straightforward to roll a sheet of paper into a cone, but in doing so we must “waste” some paper (either by cutting or overlapping the sheet). Here, we show how to create a cone from a very thin polystyrene sheet without cutting or overlapping – instead, the excess length generated by deforming the initially flat sheet is buffered locally by wrinkling. To realise this, we suspend a thin circular sheet on a soap film so that it is held by a small radial tension at the boundary. The sheet is deformed by indenting at the centre. Through wrinkling, the flat sheet deforms to a cone, and subsequent changes in the angle of the cone are accommodated by self-regulated adjustment of the wrinkles. Experimentally, the soap film used to apply the radial tension is deformed by (and hence opposes) wrinkling in the sheet. We discuss the implications of this for the wrinkle pattern selection.