Shape-matrix geometric instruments having numerous applications including, but not limited to, anti-counterfeiting, graphical passwording, games, and geometry education. A shape-matrix geometric instrument is a manufacture and/or a method whose design is based on a shape-matrix that, in turn comprises a set of building blocks that are N-dimensional polytopes. Corner shapes are positioned in or near the interior corner spaces of at least ones of the shape-matrix building blocks. At least ones of the corner shapes differ from others in at least one property or aspect including, for example, geometric shape, orientation within the building block, and one or more surface finishes, such as color, shading, cross-hatching or real or apparent texture.

Shape-matrix geometric instruments having numerous applications including, but not limited to, anti-counterfeiting, graphical passwording, games, and geometry education. A shape-matrix geometric instrument is a manufacture and/or a method whose design is based on a shape-matrix that, in turn comprises a set of building blocks that are N-dimensional polytopes. Corner shapes are positioned in or near the interior corner spaces of at least ones of the shape-matrix building blocks. At least ones of the corner shapes differ from others in at least one property or aspect including, for example, geometric shape, orientation within the building block, and one or more surface finishes, such as color, shading, cross-hatching or real or apparent texture.

A computing appliance for geometric planimetry is provided, comprising a container, a vertical dividing rule, a horizontal dividing rule, a horizontal point line and a middle point line. A fluid is filled into the container, and the vertical dividing rule, the horizontal dividing rule, the horizontal point line and the middle point line are configured on a same instructing surface of the container and arranged on each of the periphery surrounding like a rectangle. When the container is vertically disposed, a fluid level is aligned with the horizontal point line. And, when the container is tilted to form at least one angle, an area of at least one geometric shape can be derived by employing the vertical dividing rule, the horizontal dividing rule, the horizontal point line and the middle point line.

A computing appliance for geometric planimetry is provided, comprising a container, a vertical dividing rule, a horizontal dividing rule, a horizontal point line and a middle point line. A fluid is filled into the container, and the vertical dividing rule, the horizontal dividing rule, the horizontal point line and the middle point line are configured on a same instructing surface of the container and arranged on each of the periphery surrounding like a rectangle. When the container is vertically disposed, a fluid level is aligned with the horizontal point line. And, when the container is tilted to form at least one angle, an area of at least one geometric shape can be derived by employing the vertical dividing rule, the horizontal dividing rule, the horizontal point line and the middle point line.

A Lynchpin structure may be combined with one or more additional Lynchpin structures to form a compound Lynchpin propulsion structure. Each Lynchpin structure may include six pentangular areas, and one or more of the pentangular areas may include a propulsion device. The propulsion may be used to propel the compound Lynchpin propulsion structure through or over various media, such as through air, across ground, on or underwater, or through or over other media. The propulsion may include avionic propulsion, ground propulsion, hydrodynamic propulsion, or other types of propulsion. A single type of propulsion device may be used within one or more of the pentangular areas, or diverse types of propulsion may be used to provide various navigational performance or multi-mode operation. Each propulsion device may also include a device to direct the propulsion, such as a single or multi-axis gimble or adjustable aerodynamic control surface.

A Lynchpin structure may be combined with one or more additional Lynchpin structures to form a compound Lynchpin propulsion structure. Each Lynchpin structure may include six pentangular areas, and one or more of the pentangular areas may include a propulsion device. The propulsion may be used to propel the compound Lynchpin propulsion structure through or over various media, such as through air, across ground, on or underwater, or through or over other media. The propulsion may include avionic propulsion, ground propulsion, hydrodynamic propulsion, or other types of propulsion. A single type of propulsion device may be used within one or more of the pentangular areas, or diverse types of propulsion may be used to provide various navigational performance or multi-mode operation. Each propulsion device may also include a device to direct the propulsion, such as a single or multi-axis gimble or adjustable aerodynamic control surface.

Shape-matrix geometric instruments having numerous applications including, but not limited to, anti-counterfeiting, graphical passwording, games, and geometry education. A shape-matrix geometric instrument is a manufacture and/or a method whose design is based on a shape-matrix that, in turn comprises a set of building blocks that are N-dimensional polytopes. Corner shapes are positioned in or near the interior corner spaces of at least ones of the shape-matrix building blocks. At least ones of the corner shapes differ from others in at least one property or aspect including, for example, geometric shape, orientation within the building block, and one or more surface finishes, such as color, shading, cross-hatching or real or apparent texture.

Shape-matrix geometric instruments having numerous applications including, but not limited to, anti-counterfeiting, graphical passwording, games, and geometry education. A shape-matrix geometric instrument is a manufacture and/or a method whose design is based on a shape-matrix that, in turn comprises a set of building blocks that are N-dimensional polytopes. Corner shapes are positioned in or near the interior corner spaces of at least ones of the shape-matrix building blocks. At least ones of the corner shapes differ from others in at least one property or aspect including, for example, geometric shape, orientation within the building block, and one or more surface finishes, such as color, shading, cross-hatching or real or apparent texture.

An educational apparatus is disclosed. The educational apparatus has one or more first sheet members including a first side and a second side, the first side and the second side each having first markings indicating the same first surface area, and one or more first attachment portions disposed along some or substantially all of a perimeter of the one or more first sheet members. The educational apparatus also has one or more second sheet members including a first side and a second side, the first side and the second side each having second markings indicating the same second surface area, and one or more second attachment portions disposed along some or substantially all of a perimeter of the one or more second sheet members.

An educational apparatus is disclosed. The educational apparatus has one or more first sheet members including a first side and a second side, the first side and the second side each having first markings indicating the same first surface area, and one or more first attachment portions disposed along some or substantially all of a perimeter of the one or more first sheet members. The educational apparatus also has one or more second sheet members including a first side and a second side, the first side and the second side each having second markings indicating the same second surface area, and one or more second attachment portions disposed along some or substantially all of a perimeter of the one or more second sheet members.