ISSN:
1572-9168
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract We exhibit a dissection, with one degree of freedom, of an arbitrary orthoscheme in Euclidean, spherical or hyperbolic d-space into d+1 orthoschemes (Section 2); this can be interpreted as a set of relations in the scissors congruence group or, weaker, as a set of functional equations for the volume. Besides special cases where the dissection is into mutually congruent parts, we obtain, in the spherical case and for a special value of the parameter, scissors congruence formulae similar to Schläfli's period formulae for the spherical orthoscheme volume (see Section 5). In Section 6 we use the dissection to explain the structure of the volume formula for asymptotic hyperbolic 3-orthoschemes due to Lobachevsky. Finally, in Section 7, by exploiting symmetries, we show that two systems of special volume relations of Schläfli (in spherical d-space) and Coxeter (for all three geometries in dimension 3) hold even on the level of dissection. In particular, it seems that all the presently known exact values for the volume of special spherical 3-simplexes hold, independently of Schläfli's differential formula, as consequences of scissors congruence relations.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00183080
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