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October 1[edit]

Question about volumes of a cube section.[edit]

Let a cube be divided into three pieces by two parallel planes, each of which go through exactly three vertices. (Think of a cube with a vertex at the top, a vertex at the bottom and two horizontal planes) What fraction of the cube is above the top plane, between the planes and below the bottom plane (obviously the first and last answer are the same). I think this can be done by calculating the volume of the pyramid above the top plane with edge of 1 unit, the height of the pyramid is 1/3 the solid diagonal or sqrt(3)/3 and the base of the pyramid is a triangle made up of face diagonals, etc. Is this the way to do it, or is there a clean fast way (preferably one that could be extended to do the equivalent slices of a hypercube...Naraht (talk) 02:34, 1 October 2016 (UTC)[reply]

Are you seeking a higher dimensional version of Parallelepiped#Corresponding tetrahedron? If so, see Parallelotope. -- ToE 14:52, 1 October 2016 (UTC)[reply]

Simpler if you think of the pyramid as having a base formed by two edges of the cube and a face diagonal. The area of the pyramid's base is then a²/2 and its height is a where a is the length of one edge of the cube. Gandalf61 (talk) 15:35, 1 October 2016 (UTC)[reply]

For the n-cube with side 1, the volumes between consecutive slices are given by the Eulerian numbers divided by n!, a fact which is noted at (sequence A008292 in the OEIS). (You can also think of this as the probability that n independent variables distributed uniformly on [0, 1] have sum between k and k+1.) The correspondence is not obvious since the Euler numbers are given in terms of the number of decreases in a permutation. But I believe if you take x₀=0, then the map y_i=x_i-x_i-1 (if x_i≥x_i-1, y_i=1+x_i-x_i-1 (if x_i<x_i-1) is volume preserving and if you slice up the cube into simplices according to the order of the coordinates then it maps the slices with k decreases to the region where the sum is between k and k+1. --RDBury (talk) 00:22, 2 October 2016 (UTC)[reply]

Sorry, went on vacation, responding to all. Thinking of England, that's exactly what I was looking for for the corner. Gandalf61, that makes sense. RDBury, I'll have to look at the Eulerian numbers in greater depth. Thanx all!!!Naraht (talk) 21:19, 6 October 2016 (UTC)[reply]

Calculating Sine and Cosine using nothing but a single complex number[edit]

Someone told me that anyone can calculate the values of Sine and Cosine in degrees using nothing but a single complex number.

Define a complex number value "onedeg" as

onedeg = 1 * (Cos[1 deg] + i Sin[1 deg]) = 0.999847695156391 + 0.0174524064372835 i

Next calculate Sine[23.45 deg] and Cos[23.45 deg]

onedeg^23.45 = 0.917408 + 0.397949 i

voila! Cos[23.45 deg] = 0.917408 and Sin[23.45 deg] = 0.397949

It feels like magic, a single complex number is capable of calculating Sine and Cosine in degrees! 110.22.20.252 (talk) 13:25, 1 October 2016 (UTC)[reply]

See Euler's formula or maybe better De Moivre's formula which is based on it for more about this. Dmcq (talk) 13:57, 1 October 2016 (UTC)[reply]

You may also enjoy reading our Circle group and Root of unity articles. -- ToE 19:41, 1 October 2016 (UTC)[reply]

And how exactly do you calculate onedeg^23.45...?

You have to be able to calculate sin and cos in the first place in order to calculate this power.

So while what you say is true (for the reasons explained by Dmcq), it in no way actually helps in the computation.

At best, for a whole number of degrees, you can use the complex representation as a mnemonic to avoid having to remember the formulas for sine/cosine of sums of angles.

And of course - if it wasn't obvious by now - there's nothing special about the particular complex number you've picked. -- Meni Rosenfeld (talk) 18:41, 4 October 2016 (UTC)[reply]

Not only a "whole" number of degrees. Actually, for every (finite) M, no matter how great it is (e.g. one can choose M=10^100, that is more than the number of atoms in the whole universe), one can define: onedeg = Cos[360/M] + i Sin[360/M], this single definition being sufficient for calculating (by Binomial theorem) the values of Sine and Cosine for as many as M angles, i.e. for any angle whose value is a multiple of 360/M.

It seems that what the OP has heard was the following: For every finite number N of digits after the decimal point (e.g. N=100), one needs a single definition for calculating (by Binomial theorem) the values of Sine and Cosine for every angle whose value (e.g. 23.45) has at most N digits after the decimal point. HOTmag (talk) 10:54, 5 October 2016 (UTC)[reply]