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Trig of Another Color

In 1965, V.B. Temple, Mathematics Professor at Louisana College, gave a talk on Tetratrigonometry at a MAA meeting. Any idea what that might be?

The analogue to the right angle is the trirectangular trihedral angle (i.e. angle formed by mutually perpendicular lines such as the x-, y-, and z-axis). Starting from the six standard trig functions based on the relations of the lengths of the sides of a right triangle, Professor Temple defined six tetratrig functions based on the relations of the areas of the faces of a right tetrahedron.

From this, he was able to derive many tetratrig identities, such as the law of sines...or should it be the law of tetrasine? As that is all I know about his talk, I do have some questions:

  • How is tetratrig different from either spherical trig or what some call solid trig?
  • Is there such a thing as a tetraradian?
  • Did Carl Allendoerfer later reach the same results in his "Generalizations of Theorems about Triangles" in Mathematics Magazine (Nov. 1965)?
Though this column is motivated by the attractive name of "tetratrig," I need to note that applications of tetratrig exist. One example is found in McNelis/Blandino's chemistry article "657, A Method for Estimating Tetrahedral Bond Angles". For a second example, consult course catalog dsecriptions for the manufacturing trades, such as this course on Compound Angles.

Whatever it may have been in the hands of Prof Temple in 1965, tetratrigonometry still sounds fascinating and rolls off one's tongue!