Mathematica Laboratory
Two options:
3D printing | Google earth |
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Export as "STL" or "WRL" and submit to a printing service like Shapeways or Scupteo to see whether it would print. You do not have to print. You submit only the Mathematica file in the form "FirstNameLastName.nb" and indicate "3D print" somewhere in problem C. If the 3D printing process fails, don't worry, but tell in the project that it did not work. | Export as "3DS" and import into Sketchup (proversion is free for 8 hours) to see whether it imports. From there, you export to google earth. Indicate somewhere in problem C that the object is for google earth. Submit the Mathematica file in the form "FirstNameLastName.nb". If the Google earth export fails, don't worry but tell in the project that it did not work. |
S=Graphics3D[ Sphere[{0,0,0},1]]; Export["sphere.3ds",S,"3DS"]; Export["sphere.stl",S,"STL"]The model we built in the Mathematica workshop is here. It is called "genius" because it is the perfect bridge between a cuboid and a sphere. A three dimensional quadrature of the circle. This quadrature of the sphere is simple, sublime and spectacular. It also shows a football elevated to a trophy predicting a Harvard win on Sunday.
And here is the project built live during the workshop. And here is the video (M4V).
FAQ
- What does "original" mean?: Answer. Something the world has never seen. Taking a Moebius strip or paraboloid as a parametrized surface would not qualify since this has appeared and a simple googling reveals the Mathematica code for such objects or such objects have appeared. Taking a Moebius strip which builds a knot however would be called original even so such implementations appear "Google Moebius strip on a knot".
- Do we need to submit the STL or 3DS file? No need. Indicate in your project which options you choose and have the code there which produces the STL or 3DS file.
- The Mathematica program can be obtained here. The current version is Mathematica 8.04. During installation you will be prompted for an Activation Key. Students Faculty/Staff. Make sure to use your Harvard email address when registering. Contact me (knill@math.harvard.edu) if you plan to use Mathematica on a linux system.
- Mathematica is started like any other application on Macintoshs or PC's. On Linux, just type "mathematica" in a terminal to start the notebook version, or "math" to start the terminal version.
- Once Mathematica is running, copy paste any of the following lines into a cell, click with the mouse somewhere into the cell, then hold "Shift" and hit "Enter".
Plot[ x Sin[x],{x,-10,10}] | Graph function of one variable |
Plot3D[ Sin[x y],{x,-2,2},{y,-2,2}] | Graph function of two variables |
ParametricPlot[ {Cos[3 t],Sin[5 t]} ,{t,0,2Pi}] | Plot planar curve |
ParametricPlot3D[{Cos[t],Sin[t],t} ,{t,0,4Pi},AspectRatio->1] | Plot space curve |
ParametricPlot3D[{Cos[t] Sin[s],Sin[t] Sin[s],Cos[s]},{t,0,2Pi},{s,0,Pi}] | Parametric Surface |
SphericalPlot3D[(2+Sin[2 t] Sin[3 s]),{t,0,Pi},{s,0,2 Pi}] | Spherical Plot |
RevolutionPlot3D[{2 + Cos[t], t}, {t,0,2 Pi}] | Revolution Plot |
ContourPlot[Sin[x y],{x,-2,2},{y,-2,2} ] | Contour lines (traces) |
ContourPlot3D[x^2+2y^2-z^2,{x,-2,2},{y,-2,2},{z,-2,2}] | Implicit surface |
VectorPlot[{x-y,x+y},{x,-3,3},{y,-3,3}] | Vectorfield plot |
VectorPlot3D[{x-y,x+y,z},{x,-3,3},{y,-3,3},{z,0,1}] | Vectorfield plot 3D |
Integrate[x Sin[x], x] | Integrate symbolically |
Integrate[x y^2-z,{x,0,2},{y,0,x},{z,0,y}] | 3D Integral |
NIntegrate[Exp[-x^2],{x,0,10}] | Integrate numerically |
D[ Cos^5[x],x ] | Differentiate symbolically |
Series[Exp[x],{x,0,3} ] | Taylor series |
DSolve[ x''[t]==-x[t],x,t ] | Solution to ODE |
DSolve[{D[u[x,t],t]==D[u[x,t],x],u[x,0]==Sin[x]},u[x,t],{x,t}] | Solution to PDE |
Classify extrema:
ClassifyCriticalPoints[f_,{x_,y_}]:=Module[{X,P,H,g,d,S}, X={x,y}; P=Sort[Solve[Thread[D[f,#] & /@ X==0],X]]; H=Outer[D[f,#1,#2]&,X,X];g=H[[1,1]];d=Det[H]; S[d_,g_]:=If[d<0,"saddle",If[g>0,"minimum","maximum"]]; TableForm[{x,y,d,g,S[d,g],f} /.P,TableHeadings->{None,{x,y,"D","f_xx","Type","f"}}]] ClassifyCriticalPoints[4 x y - x^3 y - x y^3,{x,y}]Solve a Lagrange problem with 2 variables F[x_,y_]:=2x^2+4 x y; G[x_,y_]:=x^2 y; Solve[{D[F[x,y],x]==L*D[G[x,y],x],D[F[x,y],y]==L*D[G[x,y],y],G[x,y]==1},{x,y,L}]With 3 variables F[x_,y_,z_]:=2x^2+4 x y+z; G[x_,y_,z_]:=x^2 y + z; c=1; Solve[{D[F[x,y,z],x]==L*D[G[x,y,z],x], D[F[x,y,z],y]==L*D[G[x,y,z],y], D[F[x,y,z],z]==L*D[G[x,y,z],z], G[x,y,z]==c},{x,y,z,L}]With 3 variables and two constraints F[x_,y_,z_]:=z; G[x_,y_,z_]:=z^2-x^2-y^2; H[x_,y_,z_]:=4x-3y+8z; c=0; d=5; Solve[{D[F[x,y,z],x]==L*D[G[x,y,z],x] + M D[H[x,y,z],x], D[F[x,y,z],y]==L*D[G[x,y,z],y] + M D[H[x,y,z],y], D[F[x,y,z],z]==L*D[G[x,y,z],z] + M D[H[x,y,z],z], G[x,y,z]==c, H[x,y,z]==d}, {x,y,z,L,M}]Check that a function solves a PDE: f[t_,x_]:=(x/t)*Sqrt[1/t]*Exp[-x^2/(4 t)]/(1+ Sqrt[1/t] Exp[-x^2/(4 t)]); D[f[t,x],t]+f[t,x]*D[f[t,x],x]-D[f[t,x],{x,2}] Simplify[%] Chop[%] |