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# Clebsch diagonal cubic surface

## 3D model of Clebsch equation

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Commercial use is allowed, you must attribute the creator, you may remix this work and the remixed work should be made available under this license.

### Description

ALFRED CLEBSCH

Rudolf Friedrich Alfred Clebsch (19 January 1833—7 November 1872) was a German mathematician who worked with algebraic geometry and invariant theory. He collaborated with Paul Gordan on Clebsch—Gordan coefficients for spherical harmonics, which are now widely used in quantum mechanics.The Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface is a cubic surface is a well known form, all of whose 27 exceptional lines can be defined over the real numbers.

### Materials and methods

Print with support enabled.

Workflow (http://spolearninglab.com/curriculum/lessonPlans/math/modeling_equations.html)

MathMod-1.0, which is a replacement for K3DSurf, is Mathematical Modeling software for visualizing and animating parametric and implicit surfaces. It supports 3D/4D plotting and animation.

1. Download and install the appropriate application for your system from here.

2. Open MathMod. On the Mac you need to locate and open the mathmodcollection.js file

3. To start, Toggle Isosurfaces and click on one of the named surfaces. You should see the rendered image in the second window.

4. To change the render, adjust the x, y, and z min and max values then press the (Update|Add|Cut) & Run button

5. To Export a surface click on the Export menu and save the image as a Wavefront.obj file

6. To view the script, click on the script icon:

7. Deconstructing the script:

{

//The two choices for forms are "Iso3D" and "Param3D"

"Iso3D":

//if set to null the CND Equation input will not be displayed

{"Cnd": null,

//name of component or components

"Component": ["Octahedron"],

//Description

"Description": ["This is a Description"],

//formula

"Fxyz": ["abs(x)+abs(y)+abs(z)-1"],

//Name of surface

"Name": ["Octahedron"],

//xmin and max

"Xmax": ["2"],

"Xmin": ["-1"],

//ymin and max

"Ymax": ["1"],

"Ymin": ["-1"],

//z min and max

"Zmax": ["1"],

"Zmin": ["-1"]

},

}

8. To create your own, CTRL click on a shape on the left and select Add Current Model to MySelection

9. Scroll down to MySelection and toggle the view

10. Edit the script to reflect your formula.

{"Iso3D":

{"Cnd": ["x^2+y^2+z^2-1"],

"Component": ["Sphere"],

"Description": ["A sphere"],

"Fxyz": ["x^2+y^2+z^2-4^2"],

"Name": ["Sphere"],

"Xmax": [" 6"],

"Xmin": ["-6"],

"Ymax": [" 6"],

"Ymin": ["-6"],

"Zmax": [" 6"],

"Zmin": ["-6"]

}

}

11. Press Run to test

12. Here is the formula for a squared off sphere:

x^4 + y^4 + z^4 -1

It is a generalization of a sphere. Try increasing the powers to get nearer to a cube.

13. Many of the parametric surfaces and some of the IsoSurfaces appear to be solids. When you export them open them up in NetFabb. Chances are you will have to invert them.

a. In Netfabb If you see the exclamation point, CTRL +click on the the form and select Extended>Invert Part

14. If you are working with a surface, you will have to make the surface a solid before printing.

15. Open Meshmixer

16. Import your model

17. Click on Analysis and select Units/Dimensions:

18. Adjust the size of your object:

19. Click on Select

20. Click on your form to select some of it

21. Press E to extend your selection

22. From the Edit menu select Extrude:

23. Select Normal

24. Change the extrusion to .6 to 2

25. Accept the changes

26. Export as an STL

81*(x^3+y^3+z^3)-189*(x^2*y+x^2*z+y^2*x+y^2*z+z^2*x+z^2*y)+54*x*y*z+126*(x*y+x*z+y*z)-9*(x^2+y^2+z^2)-9*(x+y+z)+1

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