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RSS Feed for Wolfram Community showing any discussions in tag Geometry sorted by activePool Noodle Spikey
http://community.wolfram.com/groups/-/m/t/1356464
I made a [compound of 5 tetrahedra](http://mathworld.wolfram.com/Tetrahedron5-Compound.html). It's surprisingly sturdy.
![noodle spikey][1]
This was built with [pool noodles](https://www.amazon.com/dp/B01BY1S2US/). Noodles are 55" (about 4.5 feet) long with a 2 3/8" diameter. Cut in half these give a Length/Diameter ratio of 11.5789. How close is that to a perfect Length/Diameter ratio?
![pool noodles][2]
First, lets build up a dodecahedron with simple vertices and an edge length of 1.23607 or $\sqrt5-1$.
From that dodecahedron, find the ten tetrahedra with edge length $2 \sqrt2 $ .
Then find five disjoint tetrahedra. The following code works.
tup=Tuples[{1,-1},{3}];
gold=Table[RotateRight[{0, \[Phi], 1/\[Phi]},n],{n,0,2}];
dodec=RootReduce[Union[Join[tup,Flatten[Table[gold[[n]] tup[[m]],{n,1,3},{m,1,8}],1]]]/.\[Phi]-> GoldenRatio];
tetra=FindClique[Graph[#[[1]]\[UndirectedEdge]#[[2]]&/@Select[Subsets[dodec,{2}],Chop[2 Sqrt[2]-EuclideanDistance@@N[#]]==0&]],{4},All];
compounds=FindClique[Graph[#[[1]]\[UndirectedEdge]#[[2]]&/@Select[Subsets[tetra,{2}],Length[Intersection[#[[1]],#[[2]]]]==0&]],{5},All];
Manipulate[Graphics3D[{
Table[{{Yellow,Red,Green,Purple,Blue}[[n]],Tube[#,thickness]&/@Subsets[compounds[[1,n]],{2}]},{n,1,k}],
Table[{{Yellow,Red,Green,Purple,Blue}[[n]],Sphere[#,thickness 2 ]&/@compounds[[1,n]]},{n,1,k}]}, Boxed-> False, SphericalRegion->True,
ViewAngle-> Pi/10, ImageSize-> 650],
Row[{Control@{{k,5, "number shown"},1,5,1, ControlType->Setter },Spacer[15],
Control@{{thickness,.11, "thickness"},.08,.20,.01, Appearance-> "Labeled" }}], SaveDefinitions->True]
![Manipulate 5 tetrahedra][3]
The "perfect" Length/Diameter ratio for rigid tubes seems to be 11.8565. The half-noodle ratio is 11.5789. Since foam is forgiving, I figured that would give a tighter figure, and that turned out to be correct.
For a regular dodecahedron with edge length 1, the inradius and circumradius are 1.11351 and 1.40125.
For an edge length of 1.23607, the inradius and circumradius are 1.37638 and 1.73204.
Based on the sizes of the tetrahedra, we can find the scaling factor of 11.4021.
Height in inches is about 31 inches tall. Distance between vertices is about 14 inches.
The notebook also has a color template. And that's how to build a spikey from pool noodles.
[1]: http://community.wolfram.com//c/portal/getImageAttachment?filename=noodle5tetra.png&userId=21530
[2]: http://community.wolfram.com//c/portal/getImageAttachment?filename=noodles.png&userId=21530
[3]: http://community.wolfram.com//c/portal/getImageAttachment?filename=manipulate5tetra.png&userId=21530Ed Pegg2018-06-15T19:42:04ZGet a smooth line in Spherical 3D plot?
http://community.wolfram.com/groups/-/m/t/1355288
Can any one please help me to get a smooth line in Spherical 3D plot.
I tried to get the minima of the Log of a function which is as a ring in the 3D plot, but my problem is how to make this ring smoother.
I have attached my code.
Thanks.Ghady Almufleh2018-06-13T22:11:17ZSolve the brick problem/perfect box problem?
http://community.wolfram.com/groups/-/m/t/1295325
The problem of finding such a cuboid is also called the brick problem, diagonals problem, perfect box problem, perfect cuboid problem, or rational cuboid problem.
Not 100% how to put this so I will get strat to the point I have a solution for this problem but am not sure who to show
need help I am on facebook.
I find it very hard to write as I have Irlen syndrome and dyslexia I have tested it on graph paper and works.
I am struggling to put into word.
so the question is where is the best place to go?
I was going to post on here is this a no no?Aaron Cattell2018-03-03T05:57:52ZGenerate random points in a unit circle using a "For" loop?
http://community.wolfram.com/groups/-/m/t/1354124
I am new to Mathematica and I have a problem with the "For" statement. I am often in doubt where to place a "," or a ";".
I am stuck with the following : I want to generate random points in a unit circle with the following function :
randompunten[n_] := Module[{p = n, r, angle, lis, i},
lis = {};
For[i = 0;
r = RandomReal[],
angle = RandomReal[2 \[Pi]],
i <= p;
i++;
lis = Append[lis, Point[{r Cos[angle], r Sin[angle]]}]]]; lis]
The result is an empty lis
NB : I just found out that this is not the right way but aside of that : what am I doing wrong in the code ?
(When I do the Append statement outside the module things work fine .Chiel Geeraert2018-06-10T18:38:05ZDraw 2D Hexagon to 3D Hexagon prism, or Thick Hexagon Mesh?
http://community.wolfram.com/groups/-/m/t/1290648
So I tried this code for 2D, but I would like to have say a slab with a defined thickness not just a plane,
The code for 2D is:
h[x_, y_] :=
Polygon[Table[{Cos[2 Pi k/6] + x, Sin[2 Pi k/6] + y}, {k, 6}]];
Graphics[{EdgeForm[Opacity[.7]], LightBlue,
Table[h[3 i + 3 ((-1)^j + 1)/4, Sqrt[3]/2 j], {i, 10}, {j, 15}]}]
I also found and modified this code [Here][1] but this is a very thin sheet , I would like to be able to define a thickness to it:
Graphics3D[
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, 10}, {j,
15}]] /.
Polygon[l_] :> {Red, Polygon[l], Polygon[{1, 0} # & /@ l]} /.
Polygon[l_List] :> Polygon[top @@@ l], Boxed -> False,
Axes -> False, PlotRange -> All, Lighting -> "Neutral"]
Can you help me how to convert 2D hexagonal to 3D hexagonal slab with user defined thickness?
[1]: https://mathematica.stackexchange.com/questions/77312/hexagonal-mesh-on-a-3d-surfaceArm Mo2018-02-24T02:10:32ZGet Top View of a 'toroid helix' which rotates around its circular axis?
http://community.wolfram.com/groups/-/m/t/1355507
![enter image description here][1]
![enter image description here][2]
Consider a very thin toroid helix which rotates around its circular axis (which passes through individual turns) while it is on a 'friction-full' surface. How would then the top view of the overall motion look like?
(Thin means to neglect elastic compression/elongation effects while helix rolls)
I guess it should be rotating around the center point while the individual turns are rotating around instantaneous circular axis.
An animation would be very much appropriate, kindly somebody come up with that.
(image source - https://i.stack.imgur.com/yLq6t.jpg)
[1]: http://community.wolfram.com//c/portal/getImageAttachment?filename=unnamed.png&userId=1355442
[2]: http://community.wolfram.com//c/portal/getImageAttachment?filename=Movinghelix.gif&userId=1355442Anshuman Dwivedi2018-06-13T18:46:37ZFrom Intersecting Cylinders to Ambiguous Rings
http://community.wolfram.com/groups/-/m/t/1353471
Intrigued by the [Ambiguous Cylinder Illusion][1], of prof. Kokichi Sugihara, I started looking for explanations with the help of Wolfram Language.
[![enter image description here][2]][1]
The "ambiguous cylinder" can, for purposes of the viewing illusion, be reduced to its upper rim. We call the rim an "ambiguous ring". This ring can be seen as a circle or a square, depending on the viewpoint. Almost perpendicular, viewpoints can be achieved with the help of a mirror.
![enter image description here][3]
Looking parallel to the axis of the circular cylinder or of the square cylinder (prism) will show a circle or a square respectively.
A possible mathematical expression for this "ambiguous curve" can be obtained by investigating the intersection of a circular with a square cylinder. My Wolfram demonstrations [Intersection of Circular and Polygonal Cylinders][4] [1] and [Ambiguous Rings 1: Polygon Based][5] illustrate the different intersection curves obtained by changing the geometry of the cylinders.
For these intersection curves to form a ring, a tight fit between the square cylinder inside the circular one is required. The circumradius of a perfectly fitted square inside a circle of radius 1 is:
fittedradius4[t_] :=
Module[{n}, n = 2 Floor[4 (.5 t + \[Pi]/8)/\[Pi]] - 1;
Cos[t - (n + 1) \[Pi]/4]]
As can be seen in this simple Manipulate:
Manipulate[
ParametricPlot[
1/Sqrt[2] Sec[
1/2 ArcTan[Cot[2 (\[Theta] - \[Theta]0)]]] {Cos[\[Theta]],
Sin[\[Theta]]}, {\[Theta], 0, 2 \[Pi]},
PlotStyle -> Directive[AbsoluteThickness[4], Orange],
PlotRange -> 1.25, ImageSize -> Small, TicksStyle -> 6,
Prolog -> {{Thick, Dotted, Circle[]}, Cyan, AbsoluteThickness[4],
Line[{{-5, fittedradius4[\[Theta]0]}, {5,
fittedradius4[\[Theta]0]}}],
Line[{{-5, -fittedradius4[\[Theta]0]}, {5, \
-fittedradius4[\[Theta]0]}}], AbsolutePointSize[5], Black,
Point[{0, 0}]}],
{{\[Theta]0, 0.5, "axial rotation\nsquare cylinder"}, 0, 10 \[Pi],
ImageSize -> Tiny}, TrackedSymbols :> Manipulate]
![Fitted square][6]
Based on [1], this is the function for a polygonal ringset:
polyRingsetCF =
Compile[{{\[Theta], _Real}, {r, _Real}, {\[Theta]0, _Real}, {d, \
_Real}, {n, _Integer}, {\[Alpha], _Real}},
Module[{t},
t = Sec[2 ArcTan[Cot[1/2 n (\[Theta] - \[Theta]0)]]/
n]; {(*part1*){Cos[\[Pi]/n] Cos[\[Theta]] t,
Cos[\[Pi]/n] t Sin[\[Theta]],
Sec[\[Alpha]] Sqrt[-d^2 + r^2 +
2 d Cos[\[Pi]/n] t Sin[\[Theta]] -
Cos[\[Pi]/n]^2 t^2 Sin[\[Theta]]^2] -
Cos[\[Pi]/n] Cos[\[Theta]] t Tan[\[Alpha]]},
{(*part 2*)Cos[\[Pi]/n] Cos[\[Theta]] t,
Cos[\[Pi]/
n] t Sin[\[Theta]], -Sec[\[Alpha]] Sqrt[-d^2 + r^2 +
2 d Cos[\[Pi]/n] t Sin[\[Theta]] -
Cos[\[Pi]/n]^2 t^2 Sin[\[Theta]]^2] -
Cos[\[Pi]/n] Cos[\[Theta]] t Tan[\[Alpha]]}}]];
ParametricPlot3D[
polyRingsetCF[t, 1., 0.5, 0., 4, 0.], {t, -\[Pi], \[Pi]},
PlotStyle -> {{Red, Tube[.035]}}, PlotPoints -> 500,
PerformanceGoal -> "Quality", SphericalRegion -> True,
PlotRange -> 1.1, Background -> Lighter[Gray, 0.5], ImageSize -> 300,
PlotTheme -> "Marketing"]
![Complete rindset4][7]
If we now use fittedRadius[\[Theta]0] as the circumradius for the square cylinder, we can create closed ringsets as a function the axial rotation \[Theta]0 of the suare cylinder. This is a GIF scrolling through all possible fitted square cylinders inside a circular cylinder of radius 1 :
![Closed Rindsets4][8]
These intersection curves are composite curves (ringsets) consisting each of two separate curves (rings). We can select one of these two rings by introducing the cutoff angles (t1 and t2) in the parametric plot [2].
polyRingCF =
Compile[{{\[Theta], _Real}, {\[Theta]0, _Real}, {r, _Real}, \
{\[Alpha], _Real}, {n, _Integer}, {d, _Real}, {t1, _Real}, {t2, \
_Real}},(*t1 and t2 are the values of \[Theta] for switching between \
parts*)
Module[{t},(*selection of parts of a composite curve*)
t = Sec[2 ArcTan[Cot[1/2 n (\[Theta] - \[Theta]0)]]/n];
{Cos[\[Pi]/n] Cos[\[Theta]] t,
Cos[\[Pi]/n] Sin[\[Theta]] t,(*select part1 or part 2*)
Piecewise[{{1, \[Theta] <= t1 \[Pi] + 2 \[Theta]0 || \[Theta] >
t2 \[Pi] + 2 \[Theta]0}}, -1]*(Sec[\[Alpha]] Sqrt[-d^2 +
r^2 + 2 d Cos[\[Pi]/n] t Sin[\[Theta]] -
Cos[\[Pi]/n]^2 t^2 Sin[\[Theta]]^2]) -
Cos[\[Pi]/n] Cos[\[Theta]] t Tan[\[Alpha]]}]]
printring4 =
ParametricPlot3D[
polyRingCF[\[Theta], 0., fittedradius4[0.], 0., 4,
0., -.5, .5], {\[Theta], -\[Pi], \[Pi]}, SphericalRegion -> True,
PlotStyle -> {Green, Tube[.05]}, Boxed -> False, Axes -> False,
PlotRange -> 1.05, PlotPoints -> 100, ImageSize -> Small,
PlotTheme -> "ThickSurface"]
![Ring41][9]
This is a GIF showing the different views of this ring: among them a circle, a square, a "lemniscate", etc...
![enter image description here][10]
To test this "ambiguous ring" in a mirror setup, we use Printout3D to send the file to Sculpteo for 3D-printing:
Printout3D[printring4, "Sculpteo", RegionSize -> Quantity[5, "cm"]]
![Sculpteo ring41][11]
And when reflected in an experimental mirror setup:
![Reflected image][12]
The physical ring looks like a circle while the reflected ring looks like a square.
For those who like to print this ring themselves, [This][13] is the link to the Sculpteo file.
[1]: http://illusionoftheyear.com/2016/06/ambiguous-cylinder-illusion/
[2]: http://community.wolfram.com//c/portal/getImageAttachment?filename=ScreenShot2018-06-11at11.52.29AM.png&userId=20103
[3]: http://community.wolfram.com//c/portal/getImageAttachment?filename=mirror-eyeview.png&userId=68637
[4]: http://demonstrations.wolfram.com/IntersectionOfCircularAndPolygonalCylinders/
[5]: http://demonstrations.wolfram.com/preview.html?draft/35335/000250/AmbiguousRings1PolygonBased
[6]: http://community.wolfram.com//c/portal/getImageAttachment?filename=rotatingpoly4.png&userId=68637
[7]: http://community.wolfram.com//c/portal/getImageAttachment?filename=8124curveplot.png&userId=68637
[8]: http://community.wolfram.com//c/portal/getImageAttachment?filename=allClosedPolyRingsets4.gif&userId=68637
[9]: http://community.wolfram.com//c/portal/getImageAttachment?filename=1284communityring41.png&userId=68637
[10]: http://community.wolfram.com//c/portal/getImageAttachment?filename=9891ezgif.com-gif-maker.gif&userId=68637
[11]: http://community.wolfram.com//c/portal/getImageAttachment?filename=sculpteoring41.png&userId=68637
[12]: http://community.wolfram.com//c/portal/getImageAttachment?filename=IMG_5862.jpg&userId=68637
[13]: https://www.sculpteo.com/en/print/model-c2c6f7e7stl/7QTBvwnD?uuid=DSIBRp6xjqbATLH201iFSbErik Mahieu2018-06-08T14:44:16Z