Wikipedia:Reference desk/Archives/Mathematics/2021 January 21

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January 21

Shapes formed from your thumbs and index fingers

There are a few geometric shapes I could form from my thumbs and index fingers: the infinity symbol (and its rotated version, the digit 8), a rectangle, an isosceles triangle, and a kite. Are there any other geometric shapes you could form from your thumbs and index fingers? GeoffreyT2000 (talk) 15:10, 21 January 2021 (UTC)

There's a near infinite number of shapes you can create, depending (of course) on how approximate you want to allow your shape to be to its platonic ideal. For example, I can make my thumbs and index fingers into a rough circle, but if I count each knuckle and crease and join as a "corner" I can call this a 14-gon (tetradecagon?). I can also create a chevron or if I curve it, a heart shape. --Jayron32 16:36, 21 January 2021 (UTC)

Name for half a Tetrahedron?

If a Tetrahedron is split on a plane equidistant between the edges that don't have a vertex in common, the cross section is a square. This cuts the Tetrahedron into two pieces. Do those pieces have a name?Naraht (talk)

Wedge (geometry). (Found from Tetrahedron#Cross_section_of_regular_tetrahedron.) 116.86.4.41 (talk) 16:23, 21 January 2021 (UTC)

The cross section only forms a square for same special tetrahedra, including the regular tetrahedron. A general tetrahedron has three pairs of opposite unconnected edges, and for each pair there is a family of intersecting planes parallel to both edges; all split the tetrahedron into a pair of wedges – they don't need to bisect the other edges. The two obtuse wedges obtained by the bisecting intersection of a plane and the regular tetrahedron each have a double mirror-symmetry (two orthogonal planes of symmetry), and for the bisecting intersection they form a congruent pair – properties not enjoyed by these wedges in the general case. The triangular faces of the "regular" wedges are equilateral, and are as long as the two shorter parallel edges, while the longer edge has twice that length.  --Lambiam 21:58, 21 January 2021 (UTC)

Thanx. I should have said regular tetrahedron. The example with the regular tetrahedron in Wedge merely defines it as an obtuse wedge (I was hoping for a more specific name). It does make me wonder, of the original tetrahedron triangles, what percentage of the area ends up in the wedge triangle and what percentage ends up on the other wedge trapezoid. (or to put it another way, if the two wedges are different colors, what is the area of each color in the reassembled regular tetrahedron.Naraht (talk) 22:27, 21 January 2021 (UTC)

The original tetrahedron triangles that will get split can be decomposed into four equal smaller ones by connecting the midpoints of the edges. Then after the splitting one of the four becomes an end triangle of one wedge, and the other three, forming a trapezoid, remain together and become a side of the other wedge.  --Lambiam 00:52, 22 January 2021 (UTC)

Thanx!Naraht (talk) 04:56, 24 January 2021 (UTC)

@Lambiam: Did you mean 'parallel' instead of 'intersecting' for the family of planes parallel to a pair of unconnected edges? Let the vertices be A, B, C, D, and AB, CD the pair of edges in question. If AB and CD are parallel then the four points are coplanar and ABCD form a quadrilaterial in that plane. Otherwise, ${\overrightarrow {AB}}\times {\overrightarrow {CD}}$ is non-zero and perpendicular to any plane parallel to both AB and CD. Then the planes parallel to AB and CD are parallel to each other. The plane equidistant from both AB and CD has equation:

$X\cdot ({\overrightarrow {AB}}\times {\overrightarrow {CD}})={\frac {A+C}{2}}\cdot ({\overrightarrow {AB}}\times {\overrightarrow {CD}})$

Note that there are three planes formed this way, for the three pairs of unconnected edges, and they intersect at the centroid (A+B+C+D)/4. --RDBury (talk) 01:33, 22 January 2021 (UTC)

I was only considering nondegenerate cases. Then opposite edges AB and CD are not parallel to each other, but there are planes that are parallel to both AB and CD. These planes are then all parallel to each other. Some of these planes (properly) intersect the tetrahedron, other don't. What I meant by "family of intersecting planes parallel to both edges" was "the subset consisting of the intersecting planes among all planes that are parallel to both AB and CD".  --Lambiam 02:30, 22 January 2021 (UTC)

Now I understand. By 'intersecting' you meant intersecting the tetrahedron; I thought you meant intersecting each other. Thanks for the clarification. --RDBury (talk) 09:58, 22 January 2021 (UTC)