Methods and systems for designing and manufacturing mathematically fair n-sided dice
AssigneeUniversity of Alaska Anchorage
InventorKatherine Rawlins
Life of this patent
Estimating the term…
Abstract
Methods and systems for designing and manufacturing a mathematically fair N-sided die are disclosed. An example method can comprise selecting an irregular polyhedron to serve as a die, and determining a center of mass of the selected irregular polyhedron die. In an aspect, a size of each face of the irregular polyhedron can be selected such that a solid angle subtended by each face from the center of mass of the selected irregular polyhedron can be equal. In an aspect a system comprising an injection molding apparatus can be used to form the die
Claims20
A system for manufacturing a die comprising:
- a first injection mold with a chamber having dimensions comprising:
- an elongated double pyramid, wherein a central part of the elongated double pyramid's shape comprises a central prism formed by an s-sided polygon, wherein s is an integer greater than or equal to three;
- an upward-pointing s-cornered pyramid attached to an upper face of the prism;
- and
- a downward-pointing s-cornered pyramid attached to a lower face of the prism, wherein the height of the central prism is n times the length of a side of the central prism, and wherein n and s are real numbers such that the die, when tossed, will land with a substantially equal probability on any side;
- and
- an injection molding apparatus comprising:
- a top opening having dimensions suitable for passing a quantity of solid polymer material into a heating chamber, wherein the chamber is coupled to the top opening;
- a heating element coupled to the heating chamber, wherein the heating element causes the quantity of solid polymer material in the heating chamber to become molten polymer material;
- and
- an injection jet coupled to the heating chamber having dimensions suitable for passing a quantity of the molten polymer material through a head of the injection jet and into an injection chamber.
The system of 1, further comprising a second injection mold having a second chamber, wherein the dimensions of the second chamber comprise:
- a polyhedron with a height n from two triangular faces, each with equal side lengths m, three rhombus faces, each with a width at the midpoint l, and six quadrilateral faces;
- a top cap of the polyhedron having two top triangular faces;
- a bottom cap of the polyhedron having two bottom triangular faces;
- and
- a plurality of triangular face edges, each being joined to an edge of at least one of the six quadrilateral faces, wherein each of the six quadrilateral faces has an opposite edge at an equator of the polyhedron and edges of the six quadrilateral faces create the three rhombus faces, and wherein n, m, and l are real numbers such that the die, when tossed, will land with equal probability on any side.
The system of 1, further comprising a third injection mold having a third chamber, wherein the dimensions of the third chamber comprise:
- a double-pyramid polyhedron, wherein a first pyramid of the double-pyramid has a height l, a shared base of the double-pyramid is a regular polygon having a side length m, and a second pyramid of the double-pyramid, the second pyramid being truncated to form one additional face having a height n at the point where it is truncated and a height o, representing a height of the second pyramid before it is truncated;
- the first pyramid extended upward from the shared base to meet at an upper tip, creating at least one triangular face;
- and
- the second pyramid extended downward from the shared base towards a lower tip, terminating at edges and forming a lower polygonal face, wherein l, m, n and o are real numbers such that the die will land with equal probability on any side.
A method comprising:
- selecting a shape for an irregular polyhedral die consisting of two types of faces, a polygonal top, and a polygonal bottom, wherein one of the two types of faces is a rectangle positioned a predetermined distance from a center of mass of the irregular polyhedral die such that the predetermined distance can be expressed in terms of one or more dimensions of the irregular polyhedral die, and wherein the polygonal top and the polygonal bottom of the irregular polyhedral die each subtend a solid angle that is equal to a solid angle subtended by each face of the irregular polyhedral die;
- determining mathematically fair dimensions of the irregular polyhedral die such that a height of the polygonal top, a length of the polygonal top, a height of the polygonal bottom, a length of the polygonal bottom, the predetermined distance, and values of the subtended solid angles are such that the irregular polyhedral die, when tossed, will land with equal probability on any side;
- and
- forming, by an injection molding apparatus, the irregular polyhedral die.
Dependencies are inferred from the claim text.