Maverick Philosopher

Footnotes to Plato from the foothills of the Superstition Mountains

Kant, Spherical Triangles, and Incongruent Counterparts

Buckner demands an argument from incongruent counterparts to the ideality of space. But before we get to that, I am having trouble understanding how the 'spherical triangles' Kant mentions in the Prolegomena to Any Future Metaphysics, sec. 13,  are incongruent counterparts. Perhaps my powers of visualization are weak. Maybe someone can help me.

I understand how a hand and its mirror image are incongruent counterparts. If I hold up my right hand before a mirror what I see is a left hand.  As Kant says, "I cannot put such a hand as is seen in the glass in the place of its original; for if this is a right hand, that in the glass is a left one . . . ." (p. 13)  That is clear to me.

Now visualize a sphere and two non-plane 'spherical triangles' the common base of which is an arc of the sphere's equator. The remaining two sides of the one triangle meet at the north pole; the remaining two sides of the other at the south pole.  The two triangles are exact counterparts, equal in all such internal respects as lengths of sides, angles, etc.  They are supposed to be incongruent in that "the one cannot be put in place of the other (that is, upon the opposite hemisphere)." (ibid.)  That is not clear to me.

Imagine the southern triangle detached from the sphere and rotated through 180 degrees so that the south vertex is pointing north and the base is directly south. Now imagine the southern triangle place on top of the northern triangle.  To my geometrical intuition they are congruent!

So, as I see it, hands and gloves are chiral but Kant's spherical triangles are not.

Wikipedia:

In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be achiral.

A chiral object and its mirror image are said to be enantiomorphs. The word chirality is derived from the Greek χείρ (cheir), the hand, the most familiar chiral object; the word enantiomorph stems from the Greek ἐναντίος (enantios) 'opposite' + μορφή (morphe) 'form'.

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Bill Vallicella

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6 responses to “Kant, Spherical Triangles, and Incongruent Counterparts”

  1. Joe Odegaard Avatar
    Joe Odegaard

    The way that they are not congruent is sort of a technicality, but: Before you rotate the northern hemisphere triangle onto the southern one, label the ascending sides, (the ones which arrive at the north pole), “W” for the west one, and “E” for the east one. Now do the same for the southern spherical triangle. Then, when you rotate the northern triangle around the mid-point of its base, so that shape-wise it fits exactly over the southern triangle, you will find that the northern “W” is now above the southern “E.” The other side is swapped too, of course. Spherical triangles can have interior angles which add up to more than 180º, as well. The ones in this example have 3, 90º corners, at least as measured locally, for a total of 270º. Too much fun !

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  2. Valeriu Avatar
    Valeriu

    Just a guess: you cannot visualize the fact that the two spherical triangles are enantiomorphs because you take them as being isosceles or equilateral. Try “the moves” (rotation and translation) with two scalene triangles.

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  3. Valeriu Avatar
    Valeriu

    Of course, that means to not take the poles as vertices.

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  4. Joe Odegaard Avatar
    Joe Odegaard

    The Coriolis “Force” is opposite-handed in the Southern Hemisphere from how it is in the Northern Hemisphere as well. Rotation is a strange thing indeed.

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  5. oz the ostrich Avatar
    oz the ostrich

    Thanks to Joe for the explanation. Seems right to me.

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  6. David Brightly Avatar
    David Brightly

    Not clear to me either, Bill. Why does Kant resort to spherical triangles? 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?

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