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53

We can do this by building a regular hexagon tile and wrapping it onto a torus: hexTile[n_, m_] := With[{hex = Polygon[Table[{Cos[2 Pi k/6] + #, Sin[2 Pi k/6] + #2}, {k, 6}]] &}, Table[hex[3 i + 3 ((-1)^j + 1)/4, Sqrt[3]/2 j], {i, n}, {j, m}] /. {x_?NumericQ, y_?NumericQ} :> 2 π {x/(3 m), 2 y/(n Sqrt[3])} ] ht = ...


51

I'm going to brute force it numerically. First, let's define the function we're interested in: fun = KnotData[{3, 1}, "SpaceCurve"] Imagine that this function fun[t] describes the position of a moving point in time. The the magnitude of its velocity as a function of the time t is Sqrt[#.#] & [fun'[t]] I'm going to make an interpolating function ...


39

I propose a small modification of the parametrization for the torus that addresses issues with conformality. Try F[t_, u_, r_] := {Cos[t] (r + Cos[u + Sin[u]/r]), Sin[t] (r + Cos[u + Sin[u]/r]), Sin[u + Sin[u]/r]} instead. Next, we wish to choose suitable values for $m, n$ for a given $r$ such that the mapping of the ...


35

p1 = {Sin[t], Cos[t]}; p2 = {Cos[3 t], Sin[2 t]}; tAtMin = ArgMax[{EuclideanDistance[p1, p2]^2, 0 <= t <= 2 Pi}, t] tAtMax = ArgMin[{EuclideanDistance[p1, p2]^2, 0 <= t <= 2 Pi}, t] ParametricPlot[{p1, p2}, {t, 0, 2 Pi}, Epilog -> {PointSize[0.02], Red, Thick, Dashed, Through[{Point, Line}[{p1, p2} /. t -> tAtMax]], Darker@Green, ...


34

General remarks In General Relativity we work in a 4-dimentional Lorentzian manifold i.e. there is a metric tensor $g$ of signature $(+,-,-,-)$ or $(-,+,+,+)$. Theses signatures are mathematically equivalent and we choose the latter because of certain quite formal aspects even though there are some physically relevant reasons for choosing the former one. In ...


34

To answer your question: I don't think it's a bad or good idea to use If. It depends on how you do it. To demonstrate I'll use If combined very powerfully with Mathematica 10's ability to tell if a point is inside a specified region or not. step[position_, region_] := Module[{randomStep}, randomStep = RandomChoice[{{-1, 0}, {1, 0}, {0, -1}, {0, 1}}]; ...


33

I'm coming to the party a bit late, but here's my approach. It should work for any two polygons, including non-convex and self-intersecting ones. winding[poly_, pt_] := Round[(Total@ Mod[(# - RotateRight[#]) &@(ArcTan @@ (pt - #) & /@ poly), 2 Pi, -Pi]/2/Pi)] cross[e1_, e2_] /; (N[Det[{Subtract @@ e1, Subtract @@ e2}]] === 0.) = ...


33

Mathematica wouldn't be much helpful if one applied only formulae calculated by hand. Here we demonstrate how to calculate the desired geometric objects with the system having a definition of the curve r[t] : r[t_] := {t, t^2, t^3} now we call uT the unit tangent vector to r[t]. Since we'd like it only for real parameters we add an assumption to Simplify ...


31

A basic approach You can start with a regular density plot, restricted to the domain of x and y using RegionFunction. Then you can transform the plot to an equilateral triangle. f[p_, q_, r_] := r Sin[10 p]^2 + (1 - r) r Cos[20 q]^2; dp = DensityPlot[f[x, y, 1 - x - y], {x, 0, 1}, {y, 0, 1}, RegionFunction -> (#1 <= 1 - #2 &), ColorFunction ...


31

Edit: added Gradient -> grad[vars] option. Without this small option the code was several orders of magnitude slower. Yes, it can! Unfortunately, not automatically. There are different algorithms to do it (see special literature, e.g. Dziuk, Gerhard, and John E. Hutchinson. A finite element method for the computation of parametric minimal surfaces. ...


29

I recreated the animation on Wikipedia for those who like such things (I do). Here is the Manipulate version: R = 3; r = 1; fx[θ_, a_: 1] := (R + r) Cos[θ] - a r Cos[(R + r) θ/r]; fy[θ_, a_: 1] := (R + r) Sin[θ] - a r Sin[(R + r) θ/r]; gridlines = Table[{x, GrayLevel[0.9]}, {x, -6, 6, 0.5}]; plot[max_] := ParametricPlot[ {fx[θ], fy[θ]}, {θ, 0, ...


29

Here is a slightly different way to do it. We write a function that converts any PDE (1D/2D/3D) into discretized system matices: Needs["NDSolve`FEM`"] PDEtoMatrix[{pde_, Γ___}, u_, r__, o : OptionsPattern[NDSolve`ProcessEquations]] := Module[{ndstate, feData, sd, bcData, methodData, pdeData}, {ndstate} = NDSolve`ProcessEquations[Flatten[{pde, Γ}], ...


28

Update With the approach described in detail below and the function given by J. M. in his answer, we can additionally introduce points to the lines which vary randomly in their size. This gives the look and feel of a pen not drawing with constant thickness due to outrunning ink: ParametricPlot[{{Cos[t] (2 + 7 Cos[2 t] - Cos[4 t])/8, Sin[t]^3 (3 - 2 Cos[2 ...


28

Fixed (see below) Here's an approach: r1 = Exp[-x^3 - y] - 1 == z; r2 = y == z; We create ImplicitRegions: reg1 = ImplicitRegion[r1, {x, y, z}]; reg2 = ImplicitRegion[r2, {x, y, z}]; The intersection of these regions is the line you seek: reg = RegionIntersection[reg1, reg2]; And here is the length (note the inclusion of the range of values in ...


27

Here's one way to implement Yves's suggestion: (* arclength function *) trefarc = \[FormalS] /. First[NDSolve[ {\[FormalS]'[t] == Norm[KnotData[{3, 1}, "SpaceCurve"]'[t]], \[FormalS][0] == 0}, \[FormalS], {t, 0, 2 Pi}, Method -> "Extrapolation"]] (* length of trefoil *) end = trefarc[2 Pi]; With[{n = 25}, (* n - number of points to ...


27

There are many ways to proceed, the best one uses FrobeniusSolve : I Since we know, that a x + b == y /. Solve[{-4 a + b == 11, 16 a + b == -1}, {a, b}] // Simplify {3 x + 5 y == 43} we find FrobeniusSolve[ {3, 5}, 43] {{1, 8}, {6, 5}, {11, 2}} a bit more straightforward way : II {x, y} /. Solve[ (a x + b == y /. Solve[ {-4 a + b == 11, 16 ...


27

Edit V10! This is simple example what we can now do in real time! R = RegionUnion @@ Table[Disk[{Cos[i], Sin[i]}, .4], {i, 0, 2 Pi, Pi/6.}]; R2 = RegionBoundary@DiscretizeRegion@R; go[] := (While[r > .105, x += v; r = RegionDistance[R2, x]; Pause[.01]]; bounce[];) bounce[] := With[{normal = Normalize[x - RegionNearest[R2, x]]}, If[break, Abort[]]; ...


27

One way to approach is to calculate the basic triangle (with sides of lengths Sqrt[n], Sqrt[n+1] and 1) and then rotate it the correct amount so that they all fit together. sumAng[n_] := Sum[ArcTan[1/Sqrt[i]], {i, 1, n}]; poly[n_] := {{0, 0}, {Sqrt[n + 1], 0}, {Sqrt[n + 1], 1}}; Graphics[Table[Rotate[{Opacity[1], Hue[RandomReal[]], Polygon[poly[i]]}, ...


26

The following is based on the fact that the determinant of a matrix is equal to zero when two rows are the same. Thus, if you plug any of the points in, you get a true statement. SeedRandom[3]; pts = RandomReal[{-1, 1}, {5, 2}]; row[{x_, y_}] := {1, x, y, x*y, x^2, y^2}; eq = Det[Prepend[row /@ pts, row[{x, y}]]] == 0 (* Out: ...


24

Just wanted to update everyone that things are much simpler, - there is built in support for this: MeshFunctions -> {"ArcLength"} So for our case: Show[{ ParametricPlot3D[KnotData[{3, 1}, "SpaceCurve"][t], {t, 0, 2 Pi}, (* the trick *) Mesh -> 15, MeshFunctions -> {"ArcLength"}, (* styles *) MeshStyle -> Directive[Red, ...


24

There is an explicit formula n = 30; m = 10; f[t_, u_] := {Cos[t] (3 + Cos[u]), Sin[t] (3 + Cos[u]), Sin[u]}; Graphics3D[Polygon /@ Table[ f[(4 π)/(3 n) (Cos[π k/3] + i 3/2), (2 π)/(Sqrt[3] m) (Sin[π k/3] + (j + i/2) Sqrt[3])], {i, n}, {j, m}, {k, 6}]] % /. Polygon -> Tube I find it a bit simpler than rm -rf's solution. Here f ...


22

The idea is quite simple: Since any great circle can be parametrized as $\cos(\theta)u + \sin(\theta)v$ where $u$ and $v$ are two orthonormal vectors. One can start with $u=\{1,0,0\}, v=\{0,1,0\}$ and use RotationTransform to get out of the xy plane, then use RotationTransform again to spin around the z-axis to get all great circles with desired inclination. ...


22

Edit A new solution (no luck is needed now :P ): g[x_] := (a x + b)/(c x + 1); (*d is a scale factor or zero*) cent[x1_, x2_, x3_] := 1/3 ({x1, g[x1]} + {x2, g[x2]} + {x3, g[x3]}); fi = First@FindInstance[{ g@First@cent[x1, x2, x3] == Last@cent[x1, x2, x3], a/b != c != 0, x1 != x2 != x3, Element[{x1, x2, x3, g[x1], g[x2], g[x3],Sequence @@ ...


22

One standard way to detect circular shapes is to binarize the image and apply a distance transform: The maxima locations of the distance transform are the centers of the circles. To make this work on your ellipses, I first have to stretch them to be (roughly) circular, as @Rahul Narain suggested in a comment: img = ...


22

The kite-domino tiling is based the pinwheel tiling which is falls out of a particular decomposition of a right triangle with legs of length 1 and 2. In the code that follows, rt[{a,b,c}] represents such a right triangle and dissect indicates how such a triangle should be decomposed into smaller copies of itself. We simply iterate the dissect function on ...


22

I've decided to expand on my comment. Before I delve into the solution, let's all pause for a moment and marvel at the stereographic parametrization of a unit circle: $$\begin{pmatrix}\frac{1-t^2}{1+t^2}\\\frac{2t}{1+t^2}\end{pmatrix}$$ Sometimes also referred to as the Weierstrass substitution, it has often been used as a tool in the solution of algebraic ...


21

I prefer PolygonArea myself, but for version 7 users, there's this function by Mr.Wizard for non-intersecting polygons: polygonArea = Compile[{{v, _Real, 2}}, Block[{x, y}, {x, y} = Transpose[v]; Abs[x.RotateLeft[y] - RotateLeft[x].y]/2 ] ] It gives the same answer as the undocumented built-in (as it should).


21

p1 = Partition[{{243.8, 77.}, {467.4, 12.}, {291.8, 130.}, {476., 210.5}, {103.2, 327.}, {245.2, 110.5}, {47.4, 343.}, {87.4, 108.5}, {371., 506.5}, {384.6, 277.}, {264.6, 525.5}, {353.8, 294.5}, {113.2, 484.5}, {296., 304.5}, {459.6, 604.5}, {320.2, 466.5}, {288.2, 630.5}, {199.6, 446.5}, {138.8, 615.5}, {81.8, 410.}, {232.4, 795.}, ...


20

How about RegionPlot? RegionPlot[ { (x - 0.2)^2 + y^2 < 0.5 && 0 < x < 1 && 0 < y < 1, (x - 0.2)^2 + y^2 < 0.5 && ! (0 < x < 1 && 0 < y < 1), ! ((x - 0.2)^2 + y^2 < 0.5) && 0 < x < 1 && 0 < y < 1 }, {x, -1, 1.5}, {y, -1, 1.5}, PlotStyle -> ...


20

Here's my version. I don't know how fast/slow it is compared to the other solutions, but at least is shortish. spiral[rlist_ /; Length[rlist] >= 2] := Module[{findCentre}, findCentre[zlist_] := Module[{coslst, theta, ind, k}, k = Length[zlist] + 1; coslst = Table[ With[{dist = N@Norm[zlist[[-1]] - zlist[[l]]]}, ((rlist[[k - 1]] ...



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