Geometric Model by A. Harry Wheeler, Spherical Polar Triangles

This cut and folded tan paper model is one of several in which A. Harry Wheeler illustrated properties of polar spherical triangles. A line perpendicular to the plane of a great circle of a sphere intersects the sphere in two points called poles (for example, on the earth, the great circle of the equator has poles the North Pole and South Pole). In the model, the outer spherical triangle has vertices labeled A, B, and C. Sides a, b, and c are opposite the corresponding vertices. Vertices of the inner spherical triangle are A₂, B₂, and C₂, with sides a ₂, b₂, and c₂. A is the pole nearest A₂ of the great circle of the sphere that includes the arc B₂ C₂. B is the pole nearest B₂ of the great circle that includes the arc A₂C₂. C is the pole nearest C₂ of the great circle that includes arc A₂B₂. Also, spherical triangle A₂B₂C₂ is the polar triangle of spherical triangle ABC (A₂ is the pole nearest A of a great circle through BC and so forth).

In this model, the point C moves along the arc AC and the point B₂ along the arc B₂C₂.

The model is among those Wheeler dubbed collapsible.

Reference:

G. van Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton: Princeton University Press, 2013.

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