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The results of geometry are nothing but the natural properties of the different lines , surfaces and bodies , or of their combinations , that in great part already appeared in nature long before men existed ( radiolaria , insects , crystals , and so on ) [ 1935 , page 393 ] . We will prove a combinatorial version of this theorem and discuss some related results from geometry . There are many topics that an amateur can readily take up and even find some new results ; geometry is a particularly rich area for these . For example , the TIMSS results for geometry at grade 8 show that 24 nations scored significantly higher than U.S. students and only four nations scored significantly lower . One of the most basic results of geometry is that if you have a right triangle in which the other angles are 45 degrees , then you know that the base of the triangle has exactly the same length as the height of the triangle . Much work was done along these lines by Hamilton himself and others , producing new results about geometry in any number of dimensions . However , your book on clocks refers to a frictional effect , one that results from geometry .