Over the last 24 hours I have been obsessing over Kant's spherical triangles. He claims that they are incongruent counterparts. Now I understand how a hand and its mirror image are incongruent counterparts. (A right hand's mirror image is a left hand.) But it is not clear to me how Kant's spherical triangles are incongruent counterparts. Supplement the above diagram with a second lower triangle that shares its base (an arc of the equator) with that of the upper triangle and whose sides are two arcs whose vertex is the south pole.
David Brightly's comment is the best I received in the earlier thread. (He works in Info Tech and I believe he has an advanced degree in mathematics.) He writes,
Not clear to me either, Bill. Why does Kant resort to spherical triangles? [To show the existence of incongruent counterparts.] Consider first two right triangles in the plane with vertices (0,0), (3,0), (0,4) in triangle A and (0,0), (3,0), (0,-4) in B. In plane geometry A and B are considered congruent, not by translation or rotation in the plane but rotation out of the plane ('flipping') with their shared edge as axis. Now think of these triangles on the sphere with edges of length 3 along the equator and those of length 4 on a meridian. The lower triangle cannot be flipped into congruence with the upper—it curves 'the wrong way'. Congruence on the sphere is more restrictive than congruence in the plane. But they are mirror images of one another in the equatorial plane. Likewise, Kant's isosceles triangles cannot be flipped into registration. Has he just overlooked that they can be slid on the sphere into alignment?
As Brightly quite rightly points out, "The lower triangle cannot be flipped into congruence with the upper — it curves 'the wrong way'." That was clear to me all along. My thought was that if you rotate the lower triangle through 180 degrees so that its southern vertex points north, it would fit right over the upper triangle. I think that is what David means when he writes, "they can be slid on the sphere into alignment."
In other words, the lower triangle needn't be rotated off the surface of the sphere with the axis of rotation being the common base, it suffices to slide the triangles into alignment and thus into congruence along the surface of the sphere.
Therefore: Kant's spherical triangles are not incongruent counterparts or enantiomorphs.
Now David, have I understood you? I am not a mathematician and I might be making a mistake.
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