Abstract: Puzzle kits include a first puzzle and a second puzzle, each of which is formed of a plurality of polyhedral modules, or polyhedrons, connected by hinges in a continuous loop. Each polyhedron includes four faces, six edges, and at least one magnet disposed adjacent to at least one face. Magnetically stabilized assemblies of the first puzzle and the second puzzle form at least a convex polyhedrons, such as a cube.

Abstract: A graph display method, wherein a first graph corresponding to a first mathematical expression is displayed on a display, and when an operation to input, as a modified expression of the first mathematical expression, a third mathematical expression having a notation different from that of the first mathematical expression is accepted, and a third graph corresponding to the third mathematical expression is displayed on the display, the third graph is displayed on the display in a way of displaying, the way being different depending on whether a condition that the first mathematical expression is mathematically identical to the third mathematical expression is satisfied or not.

Abstract: A game system may comprise a board comprising a plurality of pegs extending from the board, wherein the pegs may be disposed in rows; at least one deflector configured to be coupled to at least two pegs aligned with one another; at least one ball configured to move on the board between and/or along the pegs and deflector(s) from a start end to a goal end of the board; at least one goal configured to be coupled to at least two pegs in a goal row of the rows of pegs, wherein the goal is configured to receive a ball; and a plurality of play selectors which determine a play for each player during a turn of a game. The goal of the game may be to get a ball into a goal by navigating across the board having the pegs and deflectors as obstacles to movement of the ball.

Abstract: A reflection teaching aid includes a mounting board and first and second plane mirrors. The mounting board has a plurality of slots that are angularly displaced from each other. When inserting edges of the first and second plane mirrors are inserted respectively into two selected ones of the slots, and when an object is disposed between reflective surfaces of the first and second plane mirrors, a selected one of included angles is formed between the reflective surfaces of the first and second plane mirrors, thereby permitting an object placed between the reflective surfaces of the first and second plane mirrors to generate a corresponding number of reflected images of the object.

Abstract: A folding book moves between a collapsed configuration and an expanded configuration. The book includes a book body and a plurality of layers forming the book body. Each layer moves between a folded configuration and an unfolded configuration. The folding book moves from the collapsed configuration to the expanded configuration by moving each layer of the plurality of layers from the folded configuration to the unfolded configuration in a predetermined order.

Abstract: A puzzle includes a puzzle board having a plurality of cutouts, each cutout having a different geometric shape; a plurality of puzzle pieces adapted to fit within the cutouts, at least some of the puzzle pieces comprising a first segment and a second segment that mate with one another, wherein the first segment and the second segment have a mathematical correlation to the cutout into which the puzzle piece fits. A method of assembling a puzzle is also described.

Abstract: A constructible playhouse may include interchangeable floors, ceilings, posts, and walls. The floors and ceilings may be a plurality of identical plates that may be connected together or stacked in a variety of ways using the posts. The plates may further include channel grooves in which additional accessories and add-ons may connect. The present invention may be easily assembled and disassembled for easy storage and portability.

Abstract: A solar collector (1) is provided having a cover (200), a back wall (300), and an absorber (100), the absorber being arranged between the cover (200) and the back wall (300) and defining, with the cover (200) and/or the back wall (300), a front and/or back cavity (14, 15), respectively. At least one spacer element (400) is provided, which extends from a front surface (102) of the absorber (100) to the cover (200) and/or from a back surface (103) of the absorber (100) to the back wall (300) and is connected or connectable directly to the absorber (100) and/or the cover (200). A method for manufacturing such a solar collector (1) is also provided.

Abstract: The 3D visualization kit helps visualize concepts relating to points, surfaces, planes, curves, contours, and vectors in three dimensions. The kit is an inexpensive solution provided along with accompanying materials for its use that will allow students to effectively visualize concepts in three dimensions and aid in understanding important calculations in multivariable calculus eliminating the abstraction normally associated with concepts in three dimensions.

Abstract: 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.

Abstract: The claimed calculus teaching and demonstration device has an internal volume in the shape of two perpendicular axes and a mathematical function that can be filled with a measured amount of liquid and thus used to illustrate the calculus concept of area under the curve.

Abstract: A polygonal device for teaching students who prefer kinesthetic learning methods is presented in which the polygonal device can represent two-dimensional polygons of varying shapes and sizes. One or more of the sides, or legs, of the polygonal device are extendable, and the angles of the polygonal device may be manipulated to a desired angle. The devices presented include, but are not limited to, triangles, quadrilaterals, parallelograms, rectangles, squares, trapezoids, and kites. In each of these devices, the legs of the device are non-detachable, i.e. the device does not come apart during normal manipulation and operation.

Abstract: A method of play, wherein play emphasizes storytelling and story recounting abilities, and wherein the method comprises: (a) providing a gaming device which provides for: a plurality of subject elements, wherein each subject element from the plurality of subject elements comprises a topic for a particular story; and a plurality of situational elements, wherein each situational element from the plurality of situational elements qualifies the subject element; (b) pairing a subject element from the plurality of subject elements with a situational element from the plurality of situational elements; and (c) providing a story based on the paired subject element and selected situational element.

Abstract: 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.

Abstract: A polygonal device for teaching students who prefer kinesthetic learning methods is presented in which the polygonal device can represent two-dimensional polygons of varying shapes and sizes. One or more of the sides, or legs, of the polygonal device are extendable, and the angles of the polygonal device may be manipulated to a desired angle. The devices presented include, but are not limited to, triangles, quadrilaterals, parallelograms, rectangles, squares, trapezoids, and kites. In each of these devices, the legs of the device are non-detachable, i.e. the device does not come apart during normal manipulation and operation.

Abstract: A polygonal device for teaching students who prefer kinesthetic learning methods is presented in which the polygonal device can represent two-dimensional polygons of varying shapes and sizes. One or more of the sides, or legs, of the polygonal device are extendable, and the angles of the polygonal device may be manipulated to a desired angle. The devices presented include, but are not limited to, triangles, quadrilaterals, parallelograms, rectangles, squares, trapezoids, and kites. In each of these devices, the legs of the device are non-detachable, i.e. the device does not come apart during normal manipulation and operation.

Abstract: The present invention provides a teaching aid system (10) for assisting students in applying trigonometric classroom lessons in real life applications both in indoor and outdoor settings. The teaching aid system consists of a pivoting scope (16) and an asymmetrically weighing gravity dial (18) that are pivotally mounted to a pair of support members (12). A tape measure (42) and an arithmetic electronic calculator (46) are attached on the support members (12) to enable students compute trigonometric formulations and applications of the same, when utilizing the system for real life objects either in indoor or outdoor settings. For determining dimensions of viewed objects, the student determines the distance from the viewed object using the tape measure (42), and the angles between the top and bottom of the viewed object using the scope (16) and gravity dial (18).

Abstract: A calculus teaching aid includes a substrate, a plurality of visual components configured to define a two-dimensional graphical model of the first derivative, and a plurality of symbolic labels configured to define a variety of characteristics of the graphical model. The substrate, visual components and labels are preferably magnetized for secure attachment to one another. The calculus teaching aid may be configured in a plurality of different ways to demonstrate the various aspects of the first derivative.

Abstract: A polygonal device for teaching students who prefer kinesthetic learning methods is presented in which the polygonal device can represent two-dimensional polygons of varying shapes and sizes. One or more of the sides, or legs, of the polygonal device are extendable, and the angles of the polygonal device may be manipulated to a desired angle. The devices presented include, but are not limited to, triangles, quadrilaterals, parallelograms, rectangles, squares, trapezoids, and kites. In each of these devices, the legs of the device are non-detachable, i.e. the device does not come apart during normal manipulation and operation.

Abstract: The present invention discloses an interactive digital learning system and method using multiple representations to assist in geometry proofs. The system of the present invention links to a computer and comprises a database, a problem representation unit, a proof representation unit and a visual representation unit. The database respectively links to the computer, problem representation unit, proof representation unit and visual representation unit and has a geometry proof problem containing a given condition, a prove statement, a static geometric figure and a solution. The problem representation unit displays the geometry proof problem. The solution is presented in a formal proof unit and a proof tree unit in form of short-answer questions, multi-choice questions and logic arrangement tests to train students. The visual representation unit presents the static geometric figure and a dynamic geometric figure to deepen the students' comprehension on geometry proof problem.

Abstract: Devices and processes for use in computer aided or computer navigated surgery include one or more fault interfaces interposed between an indicium and an item to be used during surgery such as a body part, tool, implant, trial or other structure or component. After the indicia have been registered into the system and surgery begun, it is sometimes the case that indicia can be inadvertently moved or dislodged in position and/or orientation relative to the body part. Fault interfaces according to various embodiments are designed to fail first, so that the indicia can be repositioned relative to the item without the need to reregister the indicia into the system relative to the item. The fault interfaces preferably include structure that allows the indicium to be repositioned relative to the item so that it does not need to be reregistered into the system. Frame attachments which can by easily manufactured at relatively low cost, are disposable, and which are manufactured for single use are disclosed.

Abstract: A convex polyhedron is formed from planar polygonal members, wherein aligned sides of adjacent abutting polygonal members meet to form edges of the polyhedron. A flexible polycarbonate plastic joining tab fits into slots in the centers of sides of adjacent polygonal members to enable adjustment of the dihedral angle between the adjacent polygonal members. A supplemental source of support such as hook and loop strips are used to fasten joined polygonal members securely in place at a desired alignment and dihedral angle. The polygonal members have mirrored interior faces which impart infinite internal reflections within an interior of the polyhedron, wherein one of the polygonal members is a two-way transparent mirror serving as a viewport, allowing a user to see into the infinitely-reflective interior of the polyhedron.

Abstract: A method for instructing mathematics includes providing a work surface to a visually impaired student; providing a plurality of movable elements for use in connection with the work surface, each of the movable elements comprising on a front surface thereof at least a portion of a visible mathematical symbol readable via eyesight, each of the movable elements further comprising Braille indicia on the front surface thereof corresponding to the at least a portion of the visible mathematical symbol, each of the moveable elements being movable to any position on the work surface; and having the student arrange a plurality of the moveable element on the work surface to form a standard mathematical expression wherein the mathematical expression comprises multiple lines set forth in a vertical arrangement.

Abstract: The systems and methods of the invention are directed to portfolio optimization and related techniques. For example, the invention provides a method for multi-objective portfolio optimization for use in investment decisions based on competing objectives and a plurality of constraints constituting a portfolio problem, the method comprising: generating an initial population of solutions of portfolio allocations; performing a first multi-objective process, based on the initial population and the competing objectives, to generate a first interim efficient frontier; performing a second multi-objective process, based on the initial population and the competing objectives, to generate a second interim efficient frontier; and fusing the first interim efficient frontier with the second interim efficient frontier to create an augmented efficient frontier for use in investment decisioning.

Abstract: A system and a method for finding integer solutions of equations whose graphs are conic sections. The system provides a physical implementation of a geometric formulation of integer solutions of conic sections. The system includes a first source of waves and an arrangement of a plurality of reflectors to provide a lattice of interference patterns of standing waves in a plane, the lattice representing intersections at integer values. The system also includes a second source of waves and a detector provided along a curve that, with the second source, defines a cone of waves, which intersects with the plane of the lattice to provide a conic section. The detector finds points of intersection of the lattice and the conic section to determine integer solutions of the conic section. The conic section may be y=N/x, in which case the integer solutions provide a factorization into integers of N.

Abstract: Disclosed are educational straws having an end portion of a defined shape and the name of said shape displayed on the exterior body of the straw thereby instilling a cognitive recognition of a sequence of letters associated with the shape which will enhance vocabulary, spelling and object spatial recognition for a plurality of defined shapes

Abstract: Provided are cubes from which composite structures may be produced. Composite structures constructed from a plurality of cubes according to the invention are useful as an instructional aid to students for enabling them to readily visualize front perspective, end perspective, and overhead perspective views of complex geometrical shapes. Visualization of such perspective views enables students to more readily produce drawings of such perspective views in paper renditions of such views, and exercises the three-dimensional thinking centers of the brain.

Abstract: The devices for teaching elementary fractional concepts each include a board having a raised periphery and central depression representing unity. Each embodiment includes a series of fractional pieces of different sizes for removable installation therein, with the number of pieces of each size having a total span equal to the span of the central depression. Thus, each piece represents the denominator of a fractional portion of the central recess. The pieces may be stacked in several layers, thus demonstrating the equivalency of different sizes of fractional pieces subtending the same area of the recess. The devices may be formed in rectangular or circular configurations, and may represent the basic abstract concept of fractions, monetary values in different systems, time and divisions thereof, etc. Separate boards may be provided for each embodiment, or inserts may be provided for placement within recesses of correspondingly configured boards.

Abstract: Methods and systems for translating models generated in one modeling environment into models that can be used in other modeling environments are disclosed. Models are created using different data formats in different modeling environments. These data formats are generally incompatible with each other. Therefore, the present invention provides a neutral data format that can store information relating to models generated in one modeling environment, and that can be used by other modeling environments to create their models. The present invention may export models created in one modeling environment into the neutral data format. The neutral data format may subsequently be imported into other modeling environments in which new models are generated using the information contained in the neutral data format. The present invention also provides animation of the newly generated models by animation clients via open animation interfaces that support multiple animation clients at a simultaneous time.

Abstract: A device that teaches the relationship between a right triangle, the length of its hypotenuse, the length of its two sides and the trigonometric functions. The device includes a horizontal and vertical ruler attached by a sliding attachment bracket. A circular plate showing 360 degrees (?) is attached to the horizontal ruler along with a pivoting ruler that can rotate 360 degrees. By sliding the vertical ruler along the horizontal ruler and revolving the pivoting ruler, the height of the vertical ruler (Y) where it intersects the pivoting ruler, the length of the horizontal ruler (X) where it intersects the vertical ruler, and the length of the of the pivoting ruler (R) where it intersects the vertical ruler can be measured. The trigonometric functions can then be calculated and plotted by their relationship with the measured values of X, Y, R and ?.

Abstract: A device that teaches the relationship between a cylinder's surface areas and volume, its diameter and radius, the top circle circumference and area, the arc length, the sector area, the volume of a slice, the front area of the slice and the side area of the slice. The device includes a transparent hollow outer half-cylinder, and a solid inner half-cylinder that can rotates around a common center for both the inner and outer half-cylinders. The outer half-cylinder has marked off units around the 180 degrees of the half-cylinder. The equations of arc length and sector area along with examples of their meanings are shown on the top half circle. The inner half-cylinder also has marked off units around the 180 degrees of the half-cylinder. The equations for volume of a slice, the front surface area of the slice, and the side surface area of the slice are shown.

Abstract: An educational toy kit to teach the alphabetic and numeric systems, the kit comprising a plurality of pieces. Each of the pieces are shaped into one of five geometic shapes, and in various sizes, from which a letter or number may be joined to form a letter of the alphabet or a number. The shapes of the pieces consist of C-shapes, I-shapes and U-shapes. The pieces may be made of wood or plastic.

Abstract: The object of this invention is to provide a block set for educational purposes. The block set features that a plurality of block units of the same color are integrated so as to build a block bar, and two or more block bars respectively of different colors are crosswisely combined to build a block bar combination, and two or more block bar combinations interconnect one another to thereby build a variety of structures having two or more colors, wherein the block bars of the same color are symmetrically disposed. The block set thus allows a user to learn a structural stability of the structures, a sense of balance, symmetry, a sense of harmony, etc. in a natural manner, through processes of building the variety of structures and repetition of those processes.

Abstract: An entertainment device comprising a plurality of rotational member divided into two groups: triangular members 10 and pentagonal members 20, a plurality of arcuate movable pieces 30 and a base for rotatably supporting said triangular and pentagonal members. The respective axes of rotation of said triangular and pentagonal members intersect at a common point representing the center of the exterior spherical surface of the device. The triangular and the pentagonal members are symmetrically disposed with regards to said center, wherein each triangular member is centrally positioned between three pentagonal members, and each pentagonal member is centrally positioned between five triangular members. Moveable pieces are disposed and slidably attached in the areas between a triangular and a corresponding neighboring pentagonal member, wherein a subsequent rotation of the triangular and pentagonal members results in changing the disposal of the moveable pieces relative to the center.

Abstract: A novel construction set particularly useful for children safely to assemble and use two-and three-dimensional structures of a wide range of sizes and shapes with flexible foam-like tubes or rods detachable to male or female elements carried by foam-like connectors, and preferably of hexaoctahedral or spherical contour, and with such novel geometric design also useful with other construction materials for other uses as well.

Abstract: I have discovered it is possible to find an indirect measurement if the vertex angle is given and the sides and the diameter are given. Using an instrument having the shape of an isosceles triangle it is possible to have a vertex angle and two congruent sides that correspond to the diameter of a circle or the side of a cube to find an indirect measurement to solve four classic problems, one is squaring the area of a circle, two is squaring the area of a sphere, three is finding the cube root of a sphere and cubing the volume, four is to double the volume of a cube.

Abstract: An polytope model includes a plurality of polyhedral elements, each element having a plurality of slanted faces and a base. The slanted faces meet at a common apex. For example, the polyhedral elements may be n-gon pyramids. In one embodiment, the polyhedral elements are identical. In alternative embodiments, the elements may define multiple distinct polyhedrons. The polyhedral elements are arranged such that the slanted faces of each element are disposed adjacent the slanted faces of other elements. The bases of the arranged polyhedral elements form the facets of a formed polyhedron. In one embodiment, the polyhedral elements are square pyramids such that the formed polyhedron is a cube wherein the faces of the cube are the bases of the square pyramids.

Abstract: The present invention provides a model and a method and apparatus for utilizing the model to simulate an imaging scenario. The model is mathematically defined by analytical basis objects and/or polygonal basis objects. Preferably, the model is a model of the human heart and thorax. Polygonal basis objects are only used to define structures in the model that experience torsion, such as certain structures in the heart that experience torsion during the cardiac cycle. The manner in which the basis objects comprising the model are transformed by scaling, translation and rotation is defined for each basis object. In the case where a basis object experiences torsion, the rotation of the basis object will change as a function of the length along the axis of the basis object about which rotation is occurring.

Abstract: Sets of tessellatable elements are disclosed, in which a relatively low number of different elements may be combined together to provide very attractive tessellating patterns. The elements may be in the form of tiles, for paving or for covering flooring or walls, for example, game pieces, including television and computer games, pieces for use in maze construction and for many other applications.

Abstract: A manipulatable beaded string toy device comprising of a plurality of interconnectable elements, each element including a number of beads disposed on a cord. When the plurality of interconnectable elements are interconnected, a junction formed comprises at least four beads, two from each cord, each of the at least four beads is in contact with at least three other beads of the junction.

Abstract: The present invention provides a variety of laminated plastic activity boards that allow the individual to visualize and have interactive exploration of conic sections and their definitions. This allows for more class time to be spent investigating the concept of conic sections opposed to teachers lecturing and the students having to memorize them. The user is able to notice patterns which develop while drawing specific sets of conic sections by forming conjectures based on the observed outcomes of the activities. Each activity board has a reusable laminated surface which allows the ink used from a marker to be visible but not permanent. The surface can be wiped clean with a moist paper towel once the activity is completed. There are a total of four activity boards which operate independently of each other and contain different concepts. The four boards teach principles relating to the circle, ellipse, parabola and hyperbola.

Abstract: A method of incorporating small graphs on to pages of text or calculations by using small adhesive-backed appliques or labels which have been pre-printed with a graphical coordinate grid. These appliques eliminate the necessity of a separate piece of graph paper or the tedium of drawing or gluing a coordinate grid directly on the report or calculation page.

Abstract: An educational device and method is provided. Comprising a first glove having a first set of digits for receiving the fingers and thumb of one hand and a second glove having a second set of digits for receiving the fingers and thumb of the other hand. A first set of indicia is provided. Located on each digit is an indicia. The indicia located on each digit on the first glove represent a first set of indicia. This first glove further includes a second set and a third set of indicia. The indicia located on each digit of the second glove represent a fourth set of indicia. This second glove includes a fifth and sixth set of indicia. The first set of indicia and the fourth set of indicia are used to learn to count while the second indicia and the fifth indicia are used to learn right from left. The fifth indicia and the sixth indicia are used to learn geometry.

Abstract: A method and apparatus for teaching a relationship of trigonometric functions between a unit circle and a rectangular coordinate system has a primary flexible member movable between a circular configuration defining the unit circle, and a straight configuration defining the rectangular coordinate system. A plurality of secondary linear members are removably coupled at attachment points along the flexible member. The linear members are movable with the flexible member between a unit circle position and a rectangular coordinate position. In the unit circle position, each linear member is oriented with respect to the unit circle to demonstrate and represent a trigonometric function. In the rectangular coordinate position, each linear member extends perpendicularly from the flexible member defining trigonometric wave forms. The flexible member and linear members physically represent the trigonometric functions and moving the members between the two systems demonstrates the relationship between them.

Abstract: A toy that may be used to make three-dimensional designs from elastic members comprises a base and at least one, and preferably two, vertical sidewalls rotatable relative to the base. The base and the at least one sidewall each have a face surface with a plurality of retaining means disposed thereon. A plurality of elastic members may be secured to the plurality of retaining means and thereby suspended between the base and the at least one sidewall to form a three-dimensional design, for example, each member may be attached at a first point along the member to a retaining means on the at least one sidewall and at a second point along the member to a retaining means on the base. The at least one sidewall or the base can be rotated to vary the length of the elastic members, thereby changing the three-dimensional design formed thereby.

Abstract: A construction kit comprising a plurality of connectable magnetic members having north and south magnetic poles on their opposite faces.

Abstract: The structures of this invention comprise a plurality of panels. Rod formations are associated with the panels along at least some edges thereof. Joining devices consisting of elongated frame members are provided with openings for removably receiving the rod formations with the associated panel edges. Connectors are utilized for receiving the respective ends of the frame members. By providing a plurality of openings in the connectors for receiving different rod ends, and by providing a plurality of such connectors, numerous temporary structural configurations can be achieved.

Abstract: The present invention relates to a toy device designed for illustrating mathematical concepts, with the toy device made from any of a variety of flexible materials. The toy device will have at least one outer edge and at least one inner edge, as well as, two cuts, so that when the toy device is played with by a child, various mathematical concepts can be illustrated.

Abstract: A three-dimensional instructional model used to exhibit and represent specific point groups used in the application of group theory to symmetry operations and molecular structure. A pair of blocks, each have a pair of spaced faces with peripheral side surfaces extending between the faces, the number of side surfaces being equal to the principal rotational axis order of the point group to be represented. The blocks are connectable to one another with a face of each of the blocks opposing. The connection between the blocks can be made using a circular shaft inserted through openings in the blocks, by pins inserted in holes in abutting opposing faces of the blocks, or by adhering the face of one block directly to the face of the other to hold them rigidly in engagement with each other. The interconnection can also be made using a rotational locking mechanism allowing the blocks to rotate with respect to one another.

Abstract: A mathematical triangle kit designed to teach children and adults mathematical and symmetry skills utilizing various sized patterned blocks. By manipulating the patterned blocks the participant learns mathematical skills, such as fractions, multiplication, trigonometry, and geometry. Additionally, by manipulating the patterned blocks the participant also learns geometric relations, such as mirror images and other symmetry relationships amongst the patterned blocks. In the preferred embodiment, the mathematical triangle kit comprises three sets of various sized and colored right triangles with 30.degree. and 60.degree. bases.