Kreisel(Spinning Top): To illustrate the shape of a spinning top as a solution set of a simple algebraic equation needs some inventive mathematics! It is, indeed, not clear mathematically, at first sight, what this equation must look like because different equations can provide similar looking forms. The image shows the equation 60(x2+y2)z4 = (60−x2−y2−z2)3. The rotation symmetry round the vertical z axis is identified by x and y occurring in the quadratic polynomial x2+y2. The third power on the right hand side is necessary to produce the two points. And as z occurs only in even powers, the surface is symmetrical with respect to the reflection in the horizontal xy plane. The parameter 60 in the equation is used for aesthetic reasons.
Eistüte (Cone): The horizontal section through Cone is a so-called rosette curve: A small wheel rolls round the interior of an annular body while a pencil fixed to the wheel is drawing a curve (like a Spirograph). Different curves are formed depending on the ratio of the two radii. The curve closes if the ratio is a rational number. In our case it is the four- leaf clover. Our cone has the advantage that four scoops of ice-cream will fit in!
Helix: The Lemniscate is the plane curve with the equation y4+z2 = y2. It results from the circle y2+z2 =1 by substituting z by z/y. Geometrically, the substitution corresponds to a distortion of the circle to a figure of eight loop. If y4+z2 = y2 is conceived as equation with three variables x, y and z (where x is hidden) then the solution set is a surface in the three-dimensional space, i. e. the cylinder above the lemniscate. The Helix equation y4+x2z2 = y2 is yielded by substituting z by xz. From the geometrical view, this construction is a kind of folding. The symmetry with respect to x and z is clearly visible. For the final formula we added the factors 2 and 6 to slightly stretch Helix. The singular locus is a crossing of straight lines. The sections of Helix with the planes x = c or z = c are lemniscates for c ≠ 0, whereas the sections y = c are hyperbola pairs.