A variety of tools for demonstrating the mathematical properties is disclosed. In one embodiment, a demonstrator uses the tools to illuminate geometric relationships between objects in two and three-dimensional space. The tools may be a stationary exhibit or collected into a portable kit. Whether portable or stationary, the tools provide an easy to use, multi-functional, and visually captivating vehicle for demonstration of geometric properties.

Collapsible structures may be formed from planar solids. The structures may be comprised of multiple planar objects hingedly connected, where each planar object may include magnetic materials (e.g., magnets, ferromagnetic metals) or electromagnetic materials. Using the magnetic or electromagnetic materials, the connected planar objects may be arranged as a single planar object with multiple layers, or may be arranged as a three-dimensional (3-D) object, where the magnetic or electromagnetic materials may be used to retain the formed 3-D object shape. Application of a current to the electromagnetic materials may cause the collapsible structure to form the 3-D object, and removal of the electric current may cause the collapsible structure to revert to a single planar object. Multiple structures may be combined to form larger structures.

Collapsible structures may be formed from planar solids. The structures may be comprised of multiple planar objects hingedly connected, where each planar object may include magnetic materials (e.g., magnets, ferromagnetic metals) or electromagnetic materials. Using the magnetic or electromagnetic materials, the connected planar objects may be arranged as a single planar object with multiple layers, or may be arranged as a three-dimensional (3-D) object, where the magnetic or electromagnetic materials may be used to retain the formed 3-D object shape. Application of a current to the electromagnetic materials may cause the collapsible structure to form the 3-D object, and removal of the electric current may cause the collapsible structure to revert to a single planar object. Multiple structures may be combined to form larger structures.

A new class of polyhedron is constructed by decorating each of the triangular facets of an icosahedron with the T vertices and connecting edges of a “Goldberg triangle.” A unique set of internal angles in each planar face of each new polyhedron is then obtained, for example by solving a system of n equations and n variables, where the equations set the dihedral angle discrepancy about different types of edge to zero, where the independent variables are a subset of the internal angles in 6 gons. Alternatively, an iterative method that solves for angles within each hexagonal ring may be solved for that nulls dihedral angle discrepancy throughout the polyhedron. The 6 gon faces in the resulting “Goldberg polyhedra” are equilateral and planar, but not equiangular, and nearly spherical.

A new class of polyhedron is constructed by decorating each of the triangular facets of an icosahedron with the T vertices and connecting edges of a “Goldberg triangle.” A unique set of internal angles in each planar face of each new polyhedron is then obtained, for example by solving a system of n equations and n variables, where the equations set the dihedral angle discrepancy about different types of edge to zero, where the independent variables are a subset of the internal angles in 6 gons. Alternatively, an iterative method that solves for angles within each hexagonal ring may be solved for that nulls dihedral angle discrepancy throughout the polyhedron. The 6 gon faces in the resulting “Goldberg polyhedra” are equilateral and planar, but not equiangular, and nearly spherical.

An information processing method includes displaying a graph on a display screen of a display, in response to a user operation of specifying at least part of the graph, displaying an icon corresponding to a numerical value which is associated with the at least part of the graph on the display screen, in response to a user operation of selecting the icon, as at least part of a mathematical expression to execute calculation using the numerical value which is associated with the icon selected, displaying the numerical value or a variable indicating the numerical value which is associated with the icon on the display screen.

A modular and scalable system configured to generate a topology platform for training, entertainment, gaming, living, working, acting, viewing, etc. The platform can have a plurality of movable columns that are extendable and retractable. In some embodiments, some of the columns can be retracted to allow users to walk on the columns while other columns can be extended to form a barrier (e.g., wall) or form an inanimate object (e.g., table). The system may be used in furniture to provide topologies for highly customized support structure designs to accommodate varying body shapes, sizes, and individual comfort levels. The system may be used to generate topologies for greenscreen film stages to provide post-production editing, compositing with actors, stuntmen, and video, etc. The platform can be configurable and re-configurable to change the objects and features of the topology and represent different or dynamic environments at any time.

A modular and scalable system configured to generate a topology platform for training, entertainment, gaming, living, working, acting, viewing, etc. The platform can have a plurality of movable columns that are extendable and retractable. In some embodiments, some of the columns can be retracted to allow users to walk on the columns while other columns can be extended to form a barrier (e.g., wall) or form an inanimate object (e.g., table). The system may be used in furniture to provide topologies for highly customized support structure designs to accommodate varying body shapes, sizes, and individual comfort levels. The system may be used to generate topologies for greenscreen film stages to provide post-production editing, compositing with actors, stuntmen, and video, etc. The platform can be configurable and re-configurable to change the objects and features of the topology and represent different or dynamic environments at any time.

An apparatus (100) for performing spherical geometry is disclosed. The apparatus (100) includes a frame (102) having a base (104) and a sphere set (106) placed on the base (104). The sphere set (106) includes an outer sphere (108) having a transparent surface and an inner sphere (110) having a surface representing a graph template, which includes azimuth and polar angle scales. The inner sphere (110) is held in a default static orientation with reference to the base (104) and is concentric with reference to the outer sphere (108) by a levitating arrangement. The outer sphere (108) is supported on the base (104) and is rotatable in any direction independent of the inner sphere (108) for determining spherical geometry measurements based on the graph template.

An apparatus (100) for performing spherical geometry is disclosed. The apparatus (100) includes a frame (102) having a base (104) and a sphere set (106) placed on the base (104). The sphere set (106) includes an outer sphere (108) having a transparent surface and an inner sphere (110) having a surface representing a graph template, which includes azimuth and polar angle scales. The inner sphere (110) is held in a default static orientation with reference to the base (104) and is concentric with reference to the outer sphere (108) by a levitating arrangement. The outer sphere (108) is supported on the base (104) and is rotatable in any direction independent of the inner sphere (108) for determining spherical geometry measurements based on the graph template.