11May 2017 by Vincent Summers No Comments

Sphere Reciprocal? Not Inside Out, But Equation Inverse?

Education, Mathematics
One way of mathematically representing a very simple sphere in 3D space is, r2 = x2 + y2 + z2 where r equals a radius of the sphere. Solving in terms of x, y, and z, we get, x = √(r2 — y2 — z2) y = √(r2 — x2 — z2) z = √(r2 — x2 — y2) A Sphere Reciprocal Now a sphere may be the most aesthetically pleasing of the simple geometric curves. So it is natural to wonder, concerning a sphere, what if…? So what if we convert the equation into an equation for a sphere reciprocal? No, not turn the sphere inside out. Rather an inverse of the equation of a sphere? What is the graph of, r2 = 1/( x2 + y2 + z2)…
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29Sep 2015 by Vincent Summers No Comments

Collapsing then Expanding the Equation for a Sphere

Mathematics
[caption id="attachment_8626" align="alignright" width="480"] How simple is a sphere?[/caption] Equation for the Simplest Sphere The equation for a sphere with its center at the origin is: x2 + y2 + z2 = c2 Where c is a positive constant. For simplicity, let's choose a positive constant, k, such that k = c2. Equation for a Circle by Collapsing a Sphere Collapsing it in one dimension generates the equation of one of three circles: x2 + y2 = k x2 + z2 = k y2 + z2 = k Equation for a Point by Collapsing a Circle Collapsing the three circles in one dimension generates two equations representing precisely two points for each of them: For x2 + y2 = k, x²2 = k y2 = k For x2 + z2…
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28Nov 2014 by Vincent Summers No Comments

Surface Area and Adsorption

Mathematics
[caption id="attachment_6504" align="alignright" width="480"] Activated Carbon - CC-2.5 by Ravedave[/caption] Surface area? What's that? And how does it affect physical properties? There are two similar words in the English language: absorption and adsorption. While they are related, they are at the same time distinctly separate. Absorption, simply put, is sucking into the interior or volume of something. Water, for instance, is sucked into the volume of a sponge. The water is held throughout the sponge. Adsorption Adsorption is a surface phenomenon. A substance that is adsorbed is adsorbed onto the surface. It does not enter into the interior or volume of the adsorbing agent. The difference in these physical processes determines the most efficient form the absorbing or adsorbing agent should assume. [caption id="attachment_15070" align="alignright" width="340"] Zeolite Materials for Methane…
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28Nov 2014 by Vincent Summers 3 Comments

Most Efficient Shape for Holding Liquids

Mathematics
[caption id="attachment_6496" align="alignright" width="440"] Spherical Bottle.[/caption] We store liquids in a bottle. So what is the most efficient shape that uses the least glass to store the most liquid? The volume of a sphere divided by its surface area represents the greatest ratio possible of any geometrical object. We want to use the least material to construct the vessel, while it holds the most. How shall we determine what best meets our requirements? Most Efficient Shape We determine what best meets our requirements by logic supported by mathematics. V/S (sphere) = 4/3пr3/4п€r2 = r/3 Use, for purposes of comparison and illustration, a cube, whose dimensions are "a" on a side. Then, since its surface area is the area of its six sides, V/S (cube) = a3/6a2 = a/6 Now since…
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