By Gardner M.

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1. 3. Fig. 3. 5. Chapter 3 Fig. 2. 4. 4. Fig. 4. 9. 57 The Natural Function and the Exponential Scale For m,p=0 and n odd we have the tetrahedron constant of 40000 and n=3 in fig. 5. which we show for a For m=0, p=0 and n even we have the octahedron, and we have chosen C=4000 and n=4 in fig. 6. Fig. 5. 4. 7. Fig. 6. 8. 12 +e (x+y+z)3 +e (x-y-z)3 + e (-x-y+z)3 +e (-x+y-z)3 = 4 . 1 0 6 Fig. 7. 4. I 0. Fig. 8. 1 I. The Natural Function and the Exponential Scale 59 Fig. 9. 12. We have earlier found the equation for the icosahedron from the face vectors.

7. Describe the first tower surface and the helicoid with polynoms to show that finite periodicity is useful. The equations are xy+cosz=0 and xsinz+ycosz=0. 1 a. 5 (x^2+y^2+z^2)-10, {x,-4,4 }, {y,-4,4 }, {z,-4,4 },MaxRecurs ion-> 2,P lotPoints>{ {5,3},{5,3},{5,3}, Boxed->False,Axes->True,AxesLabel-> {x,y,z} ] b. ContourPlot3D[Cosh[x]+Cosh[y]+Cosh[z]-10, {x,-4,4}, {y,-4,4}, {z,-4,4}, MaxRecursion->2,PlotPoints-> { {5,3 }, {5,3 }, {5,3 } },Boxed->False, Axes->True,AxesLabel-> {x,y,z} ] Fig. 1. a.

5 Optimal Organisation and Higher Exponentials In chemistry the building block principle is common, but by no means unique for just chemistry. In astronomy it is called clustering of stars, growing to more complex structures; in biology for the building of proteins, DNA, evolution, life itself; and it exists in society as well indeed. The common dividend is the popular word hierarchy (in church), or rank (in army) or optimal organisation (science). We will show the mathematical machinery to use for the description of such structures, and also for their growth.

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A Die of Another Color by Gardner M.

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