No direct generalisation of the Chapple equation exists for tetrahedra: for example, if r << R then d may take any value between 0 and R - r , via contemplation of needle disphenoids with one pair of short edges versus tall pyramids with narrow base. On the other hand, if R = 3 r then only d = 0 is possible, and the tetrahedron is regular; so some nontrivial inequality f(R, r, d) >= 0 must operate (besides the weak, trivial R >= r + d ). Donning thoroughly inadequate protective gear, I clambered into the cage with Maple for the first of several increasingly protracted bouts of numerical exploration and gruesome algebra, in the course of which it was established to my considerable dissatisfaction that --- Monte-Carlo trials yielded a scatter plot with a well-defined boundary f = 0 ; however (probably due to ill-conditioning) it did not appear feasible to approximate coefficients of putative polynomial f via SVD (least squares). And available CAS's, along with my own geometric skills, fall well short of hacking polynomials to express R,r,d,f in terms of the tetrahedron shape modulo isometry, in the form of its six edge-lengths. Rescue from the quicksands of laborious computation comes unexpectedly in the ethereal guise of symmetry. Consider the special case of a triangular pyramid with height h , base edge length t , both centres lying on its altitude. Via Pythagoras and similar triangles, easily 12(h r + t^2/12)^2 = (h^2 + t^2/12)t^2 , 2 h R = (h^2 + t^2/3) , R + r = d + h ; eliminating t, h yields a quartic which conveniently factorises into 3 h^2 times the nontrivial polynomial *** f_3 = (R - 3 r)(R + r) - d^2 ; *** Now consider f_3 qua function of edge lengths. Rotation about the axis leaves both spheres invariant: therefore by symmetry pyramids occupy a stationary curve of f_3 . Also it is intuitively plausible that perturbing a pyramid while fixing R,d will cause r to decrease, whence f_3 = 0 should be a local minimum, at least. And bingo --- when f_3 = 0 is superposed on the scatter data, it shows a perfect fit along the boundary! See https://www.dropbox.com/s/fo59jy5b83v4bt3/f_3_plot.png However, a proof that f_3 >= 0 everywhere remains elusive --- in fact, I don't 'ave a bleedin' clue, mate). In principle the question is expressible in terms of semi-algebraic sets, so decidable; however as noted above the associated computations, whether involving Tarsky-style resolution, Groebner bases or classical resultants, appear currently intractable. Notes: For pyramids d may take sign +, - in a solution to f_3 = 0 , corresponding respectively to squat, tall pyramids with different heights h = R + r - d but the same radii R,d . In the (improper) limit as h -> 0 , also r -> 0 and R - d -> 0 ; whence f_3 -> 0 simultaneously, so factor 3 h^2 of the quartic above may be discarded. More generally as the volume vanishes but R = 1 (say), when r, R - d -> 0 we have f_3 -> 0 also, even for asymmetric tetrahedra. Furthermore, it turns out that pyramids are not the only proper tetrahedra with f_3 = 0 : I intend to explore this more fully in a subsequent instalment. Fred Lunnon On 12/29/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Recapitulating "Euler's theorem", badly garbled in an earlier post of mine:
Relates circumradius R , inradius r , centre displacement d of triangle by R^2 - 2 R r - d^2 = 0 .
Actually due to Chapple, according to http://mathworld.wolfram.com/EulerTriangleFormula.html .
Synthetic proof at
https://en.wikipedia.org/wiki/Euler's_theorem_in_geometry#/media/File:GeometryEulerTheorem.png .
Provides foundation of Poncelet's porism for triangles in Euclidean 2-space; see http://mathworld.wolfram.com/PonceletsPorism.html .
My Scrooge's Christmas present (to myself, naturally) was to investigate the extent to which Chapple's result generalises to higher dimensions: specifically,
*** What can be established about the relation between circumradius, *** *** inradius and displacement of a tetrahedron in Euclidean 3-space? ***
I'll post more about this problem after a few days, in case other social misfits out there are motivated to tackle it themselves, or alternatively can disabuse me of any delusional claim to originality.
Incidentally, Poncelet's porism now has an entire book devoted to it, Vladimir Dragović, Milena Radnović "Integrable Billiards" Birkhäuser 2011 ; chapter 1 free online at
http://www.springer.com/cda/content/document/cda_downloaddocument/9783034800... .
Fred Lunnon