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Help with Halloween costumes

Every year our family makes costumes, and this time we are making "the world" and an "oak tree". Does anyone have any ideas for doing these in Wolfram Language?

Maybe for the world, I am thinking of projecting the map of the world onto two halves of an icosahedron. One half attached to the front of a shirt and the other half attached to the back. I am close on this part, but maybe it has become too complicated. Something Is wrong here, and this approach also involves more cutting and taping than I would prefer.

ToMapCoordinates[v_] := N[ToPolarCoordinates[v][[{3, 2}]]/Degree]
verts = Map[ToMapCoordinates,  PolyhedronData["Icosahedron", "VertexCoordinates"]];
GeoTriangle[{a_, b_, c_}] := GeoGraphics[Polygon[{a, b, c}]]
GeoTriangle[verts[[#]]] & /@  PolyhedronData["Icosahedron", "FaceIndices"]

For the oak tree, I was trying to make oak leaves and acorns using PolarPlot. Does anyone have an Oak leaf? Here is my attempt at an acorn.

PolarPlot[1.4 - CubeRoot[1 + Sin[x]], {x, 0, 2 Pi}]

acorn
POSTED BY: Todd Rowland
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POSTED BY: Todd Rowland

What I am trying to do is use Wolfram Language to print out some shapes to cut out for Halloween outfits. It makes it more fun to have some code and some math behind it.

One outfit is an oak tree, which is where the discussion on the acorn and oak leaves began,

Another is the world. I plan on printing out the PolyhedronData["Icosahedron", "NetFaces"] which I can fold along the edges and tape it together to become an icosahedron. The tricky part is projecting the world map onto the faces of the icosahedron.
POSTED BY: Todd Rowland

If you don't mind me asking, what is the reason/purpose for this 'study'? I too have an interest in this field, as well as the microscopic anatomical images of different plants for the purpose of identification of unknown plants vs. 'Authentic' plants.

I am new to this Community, so am not sure if this is an appropriate inquiry, forgive me if not.

Sidney
POSTED BY: Sidney Sudberg
POSTED BY: Udo Krause

Thanks, I am sure it will be fun. Every holiday should have a little math, and Halloween especially for anything scary (Some say math is scary, hah).

Udo, do you have a link to any Wolfram Alpha leaf shapes?

Next I need to make the world. Maybe I will try a stereographic projection onto the faces of the icosahedron instead of relying on the defaults.
POSTED BY: Todd Rowland

Halloween fly-by: 2015 TB145 will pass about 1.27 lunar distance from Earth on 31 October 2015 at about 17:00 UT.
POSTED BY: Udo Krause
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POSTED BY: Udo Krause

Thanks Udo. I guess I was expecting some sort of trig polynomial. I did try the shape you made, and tried to fit it to a simple trig polynomial but without any luck.
POSTED BY: Todd Rowland

This is going to go much easier

pic = Import[FileNameJoin[{NotebookDirectory[], "test", "oak-leaf1.jpg"}]]
Export[FileNameJoin[{NotebookDirectory[], "test", "oak-edge.png"}], EdgeDetect[ColorNegate[pic]]]

Clear[pic3, pts, sTour]
pic3 = Import[FileNameJoin[{NotebookDirectory[], "test", "oak-edge.png"}]];
pts = PixelValuePositions[pic3, 1];
sTour = FindShortestTour[pts];
Graphics[Line[pts[[Last[sTour]]]]]

giving a real graphical Oak leaf

enter image description here

if one searches for two points with the greatest position difference on sTour but the smallest euclidean distance it is possible to cut the inner part of the tour out to get the contour.
POSTED BY: Udo Krause
POSTED BY: Udo Krause
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