On Wednesday, October 4, 2017 at 3:43:26 AM UTC-5, Archimedes Plutonium wrote: > Alright, before 2018, I want significant progress on this topic, in order to start the new 6th edition. I will rename it, as TRUE Math texts, previously called Correcting Math. It will be the basis of all math textbooks used from Grade School through College. I will write some of it directly for the students. But many sections are outlines for other teachers to fill in the text. > > This last topic of the unification of conic sections, planar figures with regular polyhedra, 3rd dimension solid figures, is questionable as to whether there is even a unification possible. I am going purely on a hunch. > > But, I know for sure of a unification of Elliptic Geometry with Hyperbolic Geometry with Euclidean Geometry. I am certain there is unification in that, which follows the formula:: > > Euclidean Geometry = Elliptic Geometry unioned with Hyperbolic Geometry > > What it means is that Elliptic and Hyperbolic geometries are not independent existing geometries. They are not stand alone geometries. But rather, they occur when you have Euclidean geometry with broken symmetry. > > The finest example is a straight line in Euclidean Geometry as | > > Now, you can have a Elliptic curved line as ) and a Hyperbolic curved line as ( and when you join the two curved lines )( they cancel and yield as byproduct |. > > So if you have Euclidean Geometry, and you break-its-symmetry. What you then have is elliptic and hyperbolic geometry as byproducts. > > Old Math thinks there are 3 separate and independent geometries. New Math says there is one and only one geometry-- Euclidean which is composed of elliptic unioned hyperbolic. > > So the symmetry sounds like a key concept to unify conics with polyhedra. > > So, starting a list of plane symmetry and of axis of symmetry, here we mean reflective symmetry. > > cube and octahedron have 9 planes of symmetry > > tetrahedron and square pyramid have 6 planes of symmetry > > rectangular solid has 3 planes of symmetry > > dodecahedron and icosahedron has 15 planes of symmetry > > ---------- > > oval and parabola have 1 axis of symmetry > > hyperbola and ellipse has 2 axis of symmetry > > 3-gon, equilateral triangle has 3 axis of symmetry > > 4-gon, square, 4 axis of symmetry > > 5-gon, regular pentagon, 5 axis of symmetry > > 6-gon, hexagon, 6 axis of symmetry > > 10-gon, 10 axis of symmetry > > So if there is a unification, I suspect to unifyer is symmetry. > > Now notice a circle has infinite axis of symmetry as well as a sphere. So we have a immediate tie-in of the largest. Now for the smallest, being 1 axis of symmetry and 1 plane of symmetry. The smallest 3D, I have is the rectangular solid with 3 planes of symmetry. > > So, the first question-- is there a 3rd D object with 1 plane of symmetry. > >
Now, as soon as I hit that send key, of the above post, I had an answer for that last question.
Looking at the planar figures for 1 axis of symmetry, we have the oval and the parabola.
So, a parabola in 3rd dimension is having 2 planes of symmetry.
However, if we take the Oval of 2nd dimension and so to speak-- ovalize it in 3rd dimension, so that we have the oval shape in the xy plane, the oval shape in the xz plane, we manage to extract a 3rd D object with just 1 plane of symmetry.
This is tricky, and why I continue to harp on having hands on models, for geometry in the mind is often in error-- as we can see the Apollonius and Dandelin fool's of ellipse as conic.
I think we can only ovalize a oval in two planes the xy and the xz. We cannot ovalize an oval in all three planes of xy, xz, and yz. But build a model to be sure.
I can picture a oval in 3D, so that the oval shape cancels the other oval shape, leaving just one Plane of Symmetry.