Quasi-Spherical Orbits – by Author Bob Chester

The Most Interesting Curves You've Never Heard Of by Robert G. Chester [caption id="attachment_13083" align="alignleft" width="340"] Author Robert G. Chester[/caption] Quick, what simple rotations simultaneously generate the circle, the parabola, and the intersection of a cylinder and a sphere? Can these rotations also subsume the hippopede of Eudoxus [1], the limaçon [2], Viviani’s curve [3], rhodonea [4], the lemniscate of Gerono [5], and Fuller’s “great circle railroad tracks of energy” [6]? Quasi-Spherical Orbits, or QSOs, are the dynamic three-dimensional curves that result when a point rotates simultaneously about two or more axes. These intriguing curves provide insights and yield results in mathematics and physics alike. Viviani's Curve [caption id="attachment_13087" align="alignleft" width="133"] Rotation a[/caption] A point rotates in the right hand direction around the z-axis. The orbit is a circle in…

Mathematical Equation for a Cone

You go to the mall and request a double scoop of Rocky Road ice cream. The fellow serving says "Yes," and then he asks you asks if you would like that on a wafer or a sugar cone? Since most have eaten ice cream since childhood (unless we are dairy or otherwise intolerant), the majority of people think of a simple v-shape hollow structure as a cone. It has a top. It has a bottom. But is this the kind of geometrical shape that mathematicians think of when they refer to deriving the mathematical equation of a cone? A Mathematics Cone The cone of the mathematician bears some resemblance to that, but there are differences. The figure included with this article demonstrates that there are two v-shaped portions, not one.…

Organic Chemistry: What is Ring Strain?

[caption id="attachment_16313" align="alignright" width="440"] Cyclopropane suffers from ring strain.[/caption] Carbon ring strain? What is that? Let's begin at the beginning. The element carbon is one of a small number of elements that can bond to itself repeatedly. Compare gases such as oxygen (O2), called diatomic gases because generally only two atoms unite. Oxygen is exceptional in that it does form the triatomic molecule, ozone (O3). However, carbon can form even lengthy chains. Thus, carbon can form, not only methane (CH4), but ethane (H3C-CH3), propane (H3C-CH2-CH3), butane (H3C-CH2-CH2-CH3), pentane (H3C-CH2-CH2-CH2-CH3), and so on. Curiously, carbon can also close those chains to form rings, much as a woman can close a string of pearls about her neck using a clasp. [sc name="MidArticleAdsense"] Cyclization Although methane cannot so close, nor can ethane, the…

Geometry Stop the World – I Want to Get Off

[caption id="attachment_3951" align="alignright" width="440"] Space Time Curvature - GNU Free Documentation License Version 1.2 by Johnstone[/caption] Many of us had to take plane geometry in high school. If you remember it was, “Theorem, Proof, Theorem, Proof...” What kind of ammunition did you use for the proof of a theorem? The axioms and previously proven theorems. I know, I know. Some of you want to forget the anguish. But I want you to recall the fifth axiom... Plane Geometry - Diagram 1 Look at Diagram 1. directly below. Line a is parallel to line b. That is lines a and b don’t meet. Axiom V: Through the point where line t meets line a there is no other line that can be drawn parallel to b. Now it took about 2,000…