Re: [math-fun] 3-dimensional rhombus
Michael Kleber wrote: << Russ Cox wrote:
complete the analogy:
square : rhombus :: cube : ____ ?
Do you mean to require that the twelve edges all have the same length, or that the six faces all have the same area? Or both? Or perhaps you want all faces to be congruent? Now I'm not even sure which Gareth's "equilateral parallelepiped" would best fit -- either the first or the last, I think.
Technically there are several ways to generalize the rhombus to 3D. But the most natural and symmetric way seems to me for it to be a parallepiped all of whose edges are the same length <=> all of whose faces are rhombi <=> all of whose faces are congruent rhombi. The word for this 3D object is "rhombohedron". (It generalizes neatly to n dimensions. Not sure if there is a word for the n-dimensional object, but perhaps it ought to be "rhombotope".) All such rhombotopes can be obtained from a cube by applying an arbitrary dilatation to one main diagonal while keeping its orthogonal directions fixed, I ween. --Dan
On Monday 10 July 2006 03:06, dasimov@earthlink.net wrote:
Now I'm not even sure which Gareth's "equilateral parallelepiped" would best fit -- either the first or the last, I think. ... The word for this 3D object is "rhombohedron".
D'oh. That is of course the *correct* term for what I clunkily described as an equilateral parallelepiped. -- g
I wrote:
On Monday 10 July 2006 03:06, dasimov@earthlink.net wrote:
Now I'm not even sure which Gareth's "equilateral parallelepiped" would best fit -- either the first or the last, I think.
...
The word for this 3D object is "rhombohedron".
D'oh. That is of course the *correct* term for what I clunkily described as an equilateral parallelepiped.
It has been pointed out to me that I committed one of the mortal sins of mailing list communication: misattribution. The first three lines above were in fact written by Michael Kleber. The word "rhombohedron" came from Dan. Peccavi mea culpa, mea culpa, mea maxima culpa. -- g
Sorry to bring the discussion back to geometry instead of news group interfaces. Bug Dan Asimov wrote:
Technically there are several ways to generalize the rhombus to 3D.
But the most natural and symmetric way seems to me for it to be a parallepiped
all of whose edges are the same length
<=>
all of whose faces are rhombi
I agree with you so far, of course, but:
<=>
all of whose faces are congruent rhombi.
I don't think that's equivalent at all. Given any three vectors a,b,c of unit length, the parallepiped whose eight vertices are +-a+-b+-c has all faces rhombi of edge length 2, but it generally has three different rhombic face shapes.
The word for this 3D object is "rhombohedron".
I agree that "rhombohedron" refers to one of these two things, but I'm not sure which, and in fact I'm not even convinced that different people all agree on which it is. --Michael Kleber ps. Evidence for this last hypothesis: the top two google hits for definitions of "rhombohedron" are MathWorld, "A parallelepiped bounded by six congruent rhombi," and answers.com quoting the American Heritage Dictionary, "A prism with six faces, each a rhombus." -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.
participants (3)
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dasimov＠earthlink.net -
Gareth McCaughan -
Michael Kleber