Discussion in 'The Universal Intelligence' started by admin, Aug 28, 2015.
Consciousness and UFO's
You can use simple mathematical-geometric arguments to 'define' the shape of the earth or the topological form of anything Susan!
Geometric Circles exist, as do physical disks as cylindrical cross sections using the 2-dimensionality of an area enclosed by a circle.
Projecting the 2-dimensional circle as a plane into 3 dimensions creates a 3-dimensional sphere embedded in a 3-dimension al topological space, but also as being embedded in a 4-dimensional space, also spherical in a local coordinate system with possible global coordinates defining spheres enclosed in spheres and spheres enclosed in cylinders.
If the encompassing 3-dimensional space is cyclindrical (flat earth notion) as a possible topology, then within 3-dimensional space this cylinder (or torus or other geometric topolgy) could then again become embedded as a cylinder (or flat earth geometry) within an encompassing sphere.
This inner space being in the same dimension as the outer space, say separated by the surface area of an earth either flat or spherical would then again be encompassed in a greater geometry topolgy in the same dimension.
It is only when the embedding occurs as a lower dimension say a volume within a higher dimension, that this 'Russian Doll' repetition can be violated.
A 4-dimensional sphere or 3-ball has a 3-dimensional surface area equal to the 3-dimensional volume of a Horn Torus, encompassed within a 3-dimensional sphere.
The volume of a 3-ball is V4=½π2R4 with surface area dV4/dR = 2π2R3 = 2πRxπR2 as the volume of a basic Horn Torus in 3 dimensions (Pappus Theorem) and enclosed by a basic 3-dimensional sphere of twice the radius (2R) of a possible cyclinder (flat earth disk of radius R).
From pure geometric analysis then a 'flat earth' can always become encompassed by a spherical 'bigger' volume and limit for this embedding will be a 4-dimensional volume equal to the volume of a Horn Torus within an even bigger sphere.
This is shown in the diagram and the second diagram defines the curvature of the earth from first principles, using the Pythagorean theorem and a trigonometric analysis.
So whilst a cylindrical earth is geometrically possible in 3 space dimensions, such a flat earth would ALWAYS be embedded in a spherical 3D volumar around it, with this encompassment forming a higher dimensional boundary dividing the inner space from the outer space, just as the surface of the earth divides the inner earth from the outer earth.
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