Question of perspective

Author: Georges Quénot.

Can something be a circle for me and a square for you?

The intersection of a cone and a pyramid is a circle when seen from the apex of the cone and a square when seen from the apex of the pyramid (the axes of the cone and of the pyramid are merged and oriented in opposite directions; the cone and the pyramid are placed so that the apex of the pyramid is inside the cone and the apex of the cone is inside the pyramid).

You need a VRML viewer (e.g. FreeWRL) for seeing the wrl files and directly manipulate the object:

squareCircleLine.wrl (3D line view)
squareCircleLinePlus.wrl (3D line view + 2D square and circle views)

For the physical realization, a thickness must be given to the piece. Two shapes were designed, the v0 one with a quadrilateral section and the v1 one with a hexagonal section.

squareCircleFacet_v0.wrl ((3D facet view with thickness), v0 shape)

You can use the viewpoints and animations defined in the file.

If you do not have or do not want to install a VRML viewer:

Vue du cercle — Circle view (seen from the apex of the cone)

Vue de 3/4 côté cercle — Side / circle view

Vue de 3/4 côté carré — Side / square view

Vue du carré — Square view (seen from the apex of the pyramid)

A first prototype (v0) has been realized at fabMSTIC on January 7, 2015 (thanks to Jérôme Maisonnasse for his help). With the real object, two persons each located at the right place can simultaneously see one the circle and the other the square (one eye has to be closed because the prototype is quite small, about 4 inches).

The principle can be generalized to other couples of 2D shapes like a circle and a triangle or a square and a pentagon.

triangleCircleFacet_v1.wrl (3D facet view with thickness, v1 shape)
squareCircleFacet_v1.wrl (3D facet view with thickness, v1 shape)
pentagonCircleFacet_v1.wrl (3D facet view with thickness, v1 shape)
hexagonCircleFacet_v1.wrl (3D facet view with thickness, v1 shape)

The axes of the pyramid and of the cone do not need to be aligned.

Technical details (in French).

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Note: though I got it independently, the idea that the intersection of a cone and a pyramid can be seen either as a circle or as a square depending upon the point of view is there at least since 1963 (in French):

Le cercle carré - Étude de la perception d'une forme mouvante (par Jean Beuchet, ancien directeur du laboratoire de psychologie expérimentale de l'Université Rennes 2).