Coincidences of Centers of Edge-Incentric, or Balloon, Simplices |
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Authors: | Mowaffaq Hajja |
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Institution: | (1) Department of Mathematics, Yarmouk University, Irbid, Jordan |
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Abstract: | An edge-incentric d-simplex is defined to be a d-simplex S which admits a (d − 1)-sphere that touches all the edges of S internally. The center of such a sphere is called the edge-incenter of S and is denoted by
. Equivalently, S is edge-incentric if and only if its vertices are the centers of d + 1 (d − 1)-spheres in mutual external touch, and for this reason one may call such an S a balloon d-simplex. An orthocentric d-simplex is a d-simplex in which the altitudes are concurrent. The point of concurrence is called the orthocenter and is denoted by
. The spaces of edge-incentric and of orthocentric d-simplices have the same dimension d in the sense that a d-simplex in either space can be parametrized, up to shape, by d numbers. Edge-incentric and orthocentric tetrahedra are the first two of the four special classes of tetrahedra studied in
1, Chapter IX.B, pp. 294–333].
The degree of regularity implied by the coincidence of two or more centers of a general d-simplex is investigated in 8], where it is shown that the coincidence of the centroid
, the circumcenter
, and the incenter
does not imply much regularity. For an orthocentric d-simplex S, however, it is proved in 9] that if any two of the centers
, and
coincide, then S is regular. In this paper, the same question is addressed for edge-incentric d-simplices. Among other things, it is proved that if any three of the centers
, and
of an edge-incentric d-simplex S coincide, then S is regular, and it is also shown that none of the coincidences
, and
implies regularity (except when d ≤ 3, d ≤ 4, and d ≤ 6, respectively). In contrast with the afore-mentioned results for orthocentric d-simplices, this emphasizes once more the feeling that, regarding many important properties, orthocentric d-simplices are the true generalizations of triangles.
Several open questions are posed.
Received: June 19, 2006. |
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Keywords: | Primary 52B12 Secondary 52B15 52B11 |
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