· Portal  Help Search Members Calendar 
Welcome Guest ( Log In  Register )  Resend Validation Email 
Welcome to Dozensonline. We hope you enjoy your visit. You're currently viewing our forum as a guest. This means you are limited to certain areas of the board and there are some features you can't use. If you join our community, you'll be able to access memberonly sections, and use many memberonly features such as customizing your profile, and sending personal messages. Registration is simple, fast, and completely free. (You will be asked to confirm your email address before we sign you on.) Join our community! If you're already a member please log in to your account to access all of our features: 
Myrtone 
Posted: Oct 27 2015, 03:58 AM

Casual Member Group: Members Posts: 53 Member No.: 569 Joined: 22March 10 
I've found out about an interesting property of three of the platonic solids. If s is the edge of a platonic solid (whether a tetrahedron, a cube or octahedron) and d is the diameter of the circumscribing sphere then ^{s*s}/_{d*d} is a dozenal number. For a tetrehedron it is 0;8, 0;6 for an octahedron, and 0;4 for a cube. And the ratios are exactly these numbers, not just these numbers to one dozenal place. Only one of them, by contrast, is a decimal number, the other two having repeating decimal expansions.
Furthermore the square of the sine of the angle between faces of a tetrahedron is simply 0;Ж8, exactly, not just eight ninths to two dozenal places. That aside, both the number of corners of a tetrahedron and the number of faces (these being the same number), as well as the number of edges, are both factors of twelve, neither being factors of ten, only the former being a factor of the short hundred. Being factors of twelve, they are also factors of a gross, as are the numbers of faces, sides and edges of both a cube and octahedron, these being dual. 
jrus 
Posted: Mar 20 2016, 11:48 PM

Regular Group: Members Posts: 195 Member No.: 1,156 Joined: 23October 15 
For me, 8/9 is a clearer representation than 0;Ж8. YMMV.
More generally, if you start computing the sine squared of various angles constructed from icosahedral symmetries, you end up getting various rational combinations of 1 and the golden ratio, with 5 and 15 in the denominator. These don’t have any nice “dozenal” expansion. (e.g. the angle between two lines both passing through the center of an icosahedron and through adjacent vertices is ~63.4349°, with sine squared of 4/5.) 