Supplementary files for this post are available here: https://www.dropbox.com/s/ql9fcs5u4wefp4n/data.zip?dl=0
and consisting of dxf-files for import in Blender, blend-files with various states of solution, and mp4-file with animation.
and consisting of dxf-files for import in Blender, blend-files with various states of solution, and mp4-file with animation.
Abstract
Abstract
In Example 1 utilizes the Wolfram Language to give flat curves volume with two methods and quite different results. Example 2 uses transformed multiples of an object to form several interesting arrangements of itself.
Experiment
Experiment
Example 1: Structures
Example 1: Structures
Part 1: Curves from 2D to 3D
Part 1: Curves from 2D to 3D
In[]:=
path0=SetDirectory[NotebookDirectory[]<>"/data/"]
Out[]=
D:\DDocs\0-published\woco\ams14pub\data
A set of closed curves was plotted in 2D:
In[]:=
Clear[t];pairs=2+Cos[4t],t+πSin[8t],3+Cos[4t],t+πSin[16t],5+4Cos[4t],t+πSin[8t],3+Cos[5t],t+πSin[20t],{Cos[4t]+2,3Pi/16Cos[8t]+t};pp=(ParametricPlot[#[[1]]{Cos[#[[2]]],Sin[#[[2]]]},{t,0,2Pi}])&/@pairs
1
16
5
2
1
16
5
32
5
2
1
16
Out[]=
,
,
,
,
First and last curve were exported as dxf-file:
In[]:=
Export["c1.dxf",pp[[1]]]
Out[]=
c1.dxf
In[]:=
Export["c2.dxf",pp[[5]]]
Out[]=
c2.dxf
Structure was implemented as Block to
1) extrude a curve from above vertically upwards,
2) to set a 3D-version of this curve on top of the extruded object.
1) extrude a curve from above vertically upwards,
2) to set a 3D-version of this curve on top of the extruded object.
In[]:=
Structure[r_,phi_,h_]:=Block[{pp1,pp2,pp3,h2},pp1=ParametricPlot3D[{rCos[phi],rSin[phi],h2},{t,0,2Pi},{h2,0,h},Boxed->False,Axes->False,PlotRange->All];pp2=ParametricPlot3D[{rCos[phi],rSin[phi],13-Norm[{rCos[phi],rSin[phi]}]},{t,0,2Pi},Boxed->False,Axes->False];pp3=Show[pp1,pp2,PlotRange->All,Boxed->False,Axes->False];{pp1,pp2,pp3}]r=Cos[4t]+2;phi=t+1/16PiSin[8t];d1=Structure[r,phi,10]
Out[]=
,
,
In[]:=
Export["d1.dxf",d1[[2]]]
Out[]=
d1.dxf
In[]:=
r=Cos[4t]+2;phi=3Pi/16Cos[8t]+t;d2=Structure[r,phi,10]
Out[]=
,
,
In[]:=
Export["d2.dxf",d2[[2]]]
Out[]=
d2.dxf
The “top” curves were exported, again in dxf - format.
Part 2: Animation
Part 2: Animation
Structure implementation was adapted for an animation:
Structure2[r_,phi_,h_]:=Block[{pp1,pp2,pp3,h2},pp1=Table[ParametricPlot3D[{rCos[phi],rSin[phi],h2},{t,0,2Pi},{h2,0,h1},Boxed->False,Axes->False,PlotRange->{{-3,3},{-3,3},{-1,12}}],{h1,1,h,1}];pp2=ParametricPlot3D[{rCos[phi],rSin[phi],13-Norm[{rCos[phi],rSin[phi]}]},{t,0,2Pi},Boxed->False,Axes->False];pp3=Show[Last[pp1],pp2,PlotRange->{{-3,3},{-3,3},{-1,12}},Boxed->False,Axes->False];ListAnimate[Flatten[{pp1,pp3}]]]
r=Cos[4t]+2;phi=t+1/16PiSin[8t];Structure2[r,phi,10]
Out[]=
Example 2: Flexibility
Example 2: Flexibility
Recreation - Spiral Plot of Primes - Wolfram Language
Recreation - Spiral Plot of Primes - Wolfram Language
The first 1000 prime number transformed and visualized in a polar plot:
The arm leading to the right upper corner was to be extracted via displayed position in rphiList or by hovering over the needed points and again revealing the necessary info. The plot below zoomed in to get the inner points.
The plot below was superimposed with the chosen inner points.
The chosen points were plotted, then exported as pdf-file:
Preparation & Animation - External Apps & Wolfram Language
Preparation & Animation - External Apps & Wolfram Language
In Affinity Designer it was imported as pdf as background, then a vector curve was drawn going through above points. The curve was exported as svg-file. It was imported in Blender and scaled, beveled, modified further and finally exported as stl-file.
The stl-file was imported below:
The next steps were to scale down the imported object, rotating it stepwise around the z-axis and animating the list of objects in two variations:
References
References
Collection of animations
Collection of animations
References
References
Depth
Scene: Sets of 2D Curves modified to 3D
Arthur, Christopher. “Geometric Design, An Artful Portfolio of Mathematical Graphics.” Accessed May 23, 2022. https://wolfram.com/books/profile.cgi?id=9169.
“Wolfram Language & System Documentation Center,” May 23, 2022. https://reference.wolfram.com/language/.
Audio:
Stock Music & Sound Effects - Royalty Free Audio - Storyblocks
https://storyblocks.com/audio (accessed 2022-05-23)
MEDIA MUSIC GROUP. Evening In The Distance - Atmospheric Emotional Acoustic. Accessed May 23, 2022. SBA-346804151.
Scene: Sets of 2D Curves modified to 3D
Arthur, Christopher. “Geometric Design, An Artful Portfolio of Mathematical Graphics.” Accessed May 23, 2022. https://wolfram.com/books/profile.cgi?id=9169.
“Wolfram Language & System Documentation Center,” May 23, 2022. https://reference.wolfram.com/language/.
Audio:
Stock Music & Sound Effects - Royalty Free Audio - Storyblocks
https://storyblocks.com/audio (accessed 2022-05-23)
MEDIA MUSIC GROUP. Evening In The Distance - Atmospheric Emotional Acoustic. Accessed May 23, 2022. SBA-346804151.
Flexibility
Scene: Spiral (from prime numbers) modified to several 3D-objects
Sriraman, Bharath. Handbook of the Mathematics of the Arts and Sciences.
Accessed June 9, 2022. https://link.springer.com/book/10.1007/978-3-319-57072-3.
“Wolfram Language & System Documentation Center,” June 9, 2022. https://reference.wolfram.com/language/.
Audio:
Stock Music & Sound Effects - Royalty Free Audio - Storyblocks
https://storyblocks.com/audio (accessed 2022-06-09)
Nelson, Zak. Summer - 30 Seconds. Accessed June 9, 2022 SBA-300360195.
Scene: Spiral (from prime numbers) modified to several 3D-objects
Sriraman, Bharath. Handbook of the Mathematics of the Arts and Sciences.
Accessed June 9, 2022. https://link.springer.com/book/10.1007/978-3-319-57072-3.
“Wolfram Language & System Documentation Center,” June 9, 2022. https://reference.wolfram.com/language/.
Audio:
Stock Music & Sound Effects - Royalty Free Audio - Storyblocks
https://storyblocks.com/audio (accessed 2022-06-09)
Nelson, Zak. Summer - 30 Seconds. Accessed June 9, 2022 SBA-300360195.