Do you know the angles of a face for a 7 or 14 sided die? [closed]

Question

I need a 5 and 7 sided mathematically fair die. The 10 sided die could replace the D5 (If you're wondering the angles are nearly 52°, 128° and two 90° opposites from each other), but I couldn't find a way to make a perfect D7 or the angles for a perfect D14.

Analyzing the D10 and D8 there are 3 properties

• Have a circumsphere,
• All faces must be congruent and
• Opposites sides must be parallel

I don't like the idea for using a D8, with the 8 meaning 'Roll again' so don´t answer with that or similar answers.

Answer 1

d14

According to this MathExchange answer, the faces of any such trapezohedron will have two 90-degree angles opposite from each other. The angle \$\theta\$ at the pole vertex is given by

$$ \tan \frac{\theta}{2} = \frac{a}{b} = \sqrt{\tan{\frac{\pi}{n}} \tan{\frac{\pi}{2n}}} $$

where the die has \$2n\$ faces and \$a\$ and \$b\$ are the side lengths. For \$n = 7\$ this comes out to about 36.68 degrees, which makes the "tropical" angle about 143.32 degrees.

See also this question for d10s.

d7

Attempts at a die with 7 (rollable) faces include:

• Pentagonal prism with tuned aspect ratio, e.g. the ones by GameScience analyzed in a previous question.
• Spacing 7 points as widely as possible on a sphere and then flattening the sphere at each such point, e.g. Goodman Games' Dungeon Crawl Classics dice.
• Long dice that are unstable on their ends. Teetotums are similar.

Some of these are harder to make fair under different rolling conditions/techniques, and for the most part they do not have a level top face when laid on a table.