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Figure 1: A succession of algebraic powers is generated by a selfsimilar spiral. For equal angles of rotation, the lengths of the corresponding radii are increased to the next power.  
Figure 2: Leibniz' construction of the algebraic powers from the hanging chain
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Figure 3: An example of the three solutions to the trisection of an angle
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Figure 4: The unit of action in Gauss' complex domain.
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Figure 5: In (a) the lengths of the radii are squared as the angle of rotation doubles.
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Figure 6: Squaring a complex number.
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Figure 7: Cubing a complex number.
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Figure 8: The sin of x is zP and the cosine of x is 0P. The sine of 2x is QP' and the cosine is OP'.
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Figure 9: Variations of the sine and cosine from the squaring of a complex number.
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Figure 10: Gaussian surface for the second power.
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Figure 11: Gaussian surface for the third power.
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Figure 12: Gaussian surface for the fourth power.
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Figure 13: Combined Gaussian surfaces for algebraic equations. (a) combines the surfaces based on the variations of the sine and cosine for the second power.


(b) combines the surfaces based on the variations of the sine and cosine for the third power.


Figure 14: Roots of algebraic equations represented in a Gaussian surface. (a) is the intersection of the surfaces in 13(a) with the flat plane.


(b) is the intersection of the surfaces in 13(b) with the flat plane.


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