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Polyhedrons

 

 

    It seems natural to start a journey with regular polygons, trekking up to regular polyhedrons, and so it shall be.

    There are five regular polyhedrons per se.  (I also include the sphere as the sixth polyhedron.  Hopefully you will see why...)

    The regular polyhedrons are three dimensional solid or skeleton shaped geometric figures.

    The outside surfaces are continuous and made up of equal regular polygons.

    The regular polyhedrons are: Tetrahedron (with 4 equilateral triangle faces),  Hexahedron (with 6 square faces), Octahedron (with 8 equilateral triangle faces), Dodecahedron (with 12 regular pentagon faces), and Icosahedron (with 20 equilateral triangle faces).

    All the regular polygon face edges mate with equal adjoining like polygon edges.

    If a straight line is extended inward, perpendicular from the center point of every regular polygon face on the surface of the regular polyhedron__ they will intersect at one center-midpoint of the polyhedron.   These lines are the radii of the regular polyhedron.

    A sphere with the same radii as any regular polyhedron can be enclosed within said polyhedron, and touches all polygon face center points.  The face planes thus being tangent to said sphere.  Refer to Fig - 44.

 

    Figure 38 is the Tetrahedron... with four (4) triangle faces.

 

 

    Figure 39 is the Hexahedron... with six (6) square faces.  This is probably the most well known... our common Cube.

 

 

    Figure 40 is the Octahedron... with eight (8) triangle faces.

 

 

Figure 41 is the Dodecahedron... with twelve (12) pentagon faces.

 

 

    Figure 42 is the Icosahedron... with twenty (20) triangle faces.

 

 

    Here is the Sphere in Figure 43.  In mathematical fantasy it has a infinite number of infinitesimal circles covering the surface.  ( Similar to the fantasy of a perfect circle with an infinite number of infinitesimal sides or points on the circumference.)

    However, the math presently in use to solve for the surface area of a Sphere results in four (4) circles__  pi times the sphere radius squared, times 4, equal any sphere surface area.  Why four circles?  I do not know. These circle areas, only mathematically, have the area of four equal circles... No-one knows if or how these circles may be distributed on the surface of any sphere... It is only a mathematical numerical value of area.    However, we could think of a cube having a surface area of 24 smaller squares upon its surface without violating anything... or the tetrahedron having 16 equilateral triangles... four in each of four faces...?  Or... I am pretty sure the area of four equal circles can also be the area of 16 equal smaller circles... and etc!?

 

 

    As mentioned above, Figure 44 shows a sphere within a cube.  Any regular polyhedron can also be within a sphere as show in the right figure.  All the face regular polygon corner points will touch the inside surface of the sphere.

 

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    A little review prior to the end of this web page.  I defined the radii of all regular polygons including the circle as a straight line from the center of the figure to any side midpoint.  The diameter of any regular polygon is simply 2 radii.  Thus a diameter can be a single straight line of 2 radii, or 2 radii in different directions.

    I have shown that the circumference (perimeter) of all regular polygon each have a unique ratio of the Circumference to the Diameter... just as the unique pi ratio with the circle circumference and diameter.  By defining as such, I have shown all regular polygons and the circle use like area formulae, with only the requirement of using the correct poly pi ratio in each calculation.

    I intend to show more similarities comparing the five regular polyhedrons with the sphere.  First I will compare formulae that exist, and use the just above methods I mention for area calculations.  I will show likeness in volume formulae for regular polyhedrons and the sphere.

    Ending my web site math section is going to be what I have developed, using ratios, and a commonality of formulae that is exactly alike except for each unique ratio value that solves the single surface area of all regular polygons, the circle, the regular polyhedrons, and the sphere.

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    The following table speaks for itself.  It is easy enough to recognize the commonality of formulae as one scrolls down the area and volume columns.  Originally when doing this polyhedron work, I drew out the figures and then calculated all the various values using geometry to check the math shown against traditional math.  Here again, I have read quite a few math books, without finding anything as I am showing in the following tables.  But, there very well could be someone that did some of these findings prior to me?  I suspect I may be presenting them a bit different, I hope.  At, any rate, I did it myself, and enjoyed the endeavor, finding it interesting, and possibly of some use...?

 

 

    The following Psi Table ends the math section of my web site.  The following math is something I also have not found in books and I am pretty sure it is not what it taught in schools.  I keep thinking about our Universe and everything within it being relative.  Relative is a ratio, and can be only for the moment... yet for everyday use, many things can be put into this mode of mathematics...

    Other series of like geometric shapes will also have ratio commonalities possibly making them easier to calculate values of area and volume...