## An ultrametric geometry

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This thesis verified that metric spaces can be constructed using ultrametrics d and D, where d(x,y) = 0 if x = y and d(x,y) = (1/2) k if x not equal to y, such that x-y = 2k(a/b) for a,b relatively prime to 2, and where D(A,B)= max(d(al,bl); d(a2,b2)) for A = (al,a2) and B = (bl,b2).Assuming that a line is represented by some linear equation, a one-dimensional point was defined as an element of Q and a two-dimensional point as an element of Q x Q. There was an investigation of one-dimensional points with respect to the behavior of segments, midpoints, and distances as measured by d. The function D demonstrated the behavior of midpoints, medians, and triangles, as well as the congruence relation. The study necessitated the introduction of pseudomidpoints and pseudomedians, and an unorthodox definition of angle measurement.