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This image shows one of Friedrich Froebel's divided cube (this one divided into twenty-seven smaller cubes). Froebel's cubes were used to encourage creativity in kindergarten-age children. The children could rearrange the smaller cubes into combinations that showed life, knowledge, and beauty.

Froebel's Divided Cube (Twenty-seven Smaller Cubes)

This image shows one of Friedrich Froebel's divided cube (this one divided into twenty-seven smaller…

"The part of any solid between two planes, which may be either parallel or inclined to each other: as, the frustum of a cone ... In the figure the dotted line, c, indicates the part of the cone cut off to form the frustum, f." -Whitney, 1911

Frustum of a Cone

"The part of any solid between two planes, which may be either parallel or inclined to each other: as,…

"Isometric. Common forms dodecahedron amd trapezohedron, often in combination." — Ford, 1912

Garnet

"Isometric. Common forms dodecahedron amd trapezohedron, often in combination." — Ford, 1912

"The distance of A and B below H is indicated on the V projection. Thus if to A∧h B∧h the perpendiculars A∧h A;and;r and B∧h B∧r be drawn, A∧r B∧r will be the true length of A B." —French, 1911

Construction of Geometric Revolution

"The distance of A and B below H is indicated on the V projection. Thus if to A∧h B∧h the perpendiculars…

"The H projection of the line AB in the space is a line connecting the feet of all the perpendiculars from AB to the plane. These perpendiculars form what is known as the projecting plane. If this projecting plane to be revolved about its H trace, which is the H projection of the line, until it coincides with H, the line will be seen in its true length." —French, 1911

Revolutions into the Plane of Geometric Shapes

"The H projection of the line AB in the space is a line connecting the feet of all the perpendiculars…

"A block revolved from its first position about an axis perpendicular to H through 45 degrees, then about an axis perpendicular to P through 45 degrees until the cut face MNO is parallel to the vertical plane. To avoid confusion it is well to letter or number the corresponding points as the views are carried along." —French, 1911

Successive Revolutions of Geometric Shapes

"A block revolved from its first position about an axis perpendicular to H through 45 degrees, then…

Geometrical plane.

Geometrical Plane

Geometrical plane.

Two students drawing geometrical shapes on the chalkboard. A student is also using a scale while students sitting at desks take notes.

Geometry

Two students drawing geometrical shapes on the chalkboard. A student is also using a scale while students…

Geometrical perspective drawing.

Geometry

Geometrical perspective drawing.

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a giraffe.

Giraffe

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a giraffe.

Giraffe

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a giraffe.

Giraffe

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a giraffe.

Giraffe

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

"A piece of a parallelogram left after a similar parallelogram has been removed from a corner of it. Thus, in the figure, EFGBCD is a gnomon." -Whitney, 1911

Gnomon

"A piece of a parallelogram left after a similar parallelogram has been removed from a corner of it.…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a goose.

Goose

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a goose.

Goose

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a goose.

Goose

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a goose.

Goose

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Gypsum.

Gypsum

Gypsum.

Gypsum.

Gypsum

Gypsum.

"Monoclinic. Crystals usually tabular parallel to clinopinacoid; in diamond-shaped crystals with edges beveled by prism and pyramid faces." — Ford, 1912

Gypsum

"Monoclinic. Crystals usually tabular parallel to clinopinacoid; in diamond-shaped crystals with edges…

"Monoclinic. Crystals usually tabular parallel to clinopinacoid; in diamond-shaped crystals with edges beveled by prism and pyramid faces. Other forms rare. Sometimes twinned." — Ford, 1912

Gypsum

"Monoclinic. Crystals usually tabular parallel to clinopinacoid; in diamond-shaped crystals with edges…

"The contours of a half sphere are a series of circles, far apart near the center (top) and near together at the outside (bottom), showing that the slope of a hemisphere varies at all points, being nearly flat on top and increasing in steepness near bottom." — Moss, 1914

Contours of a half sphere

"The contours of a half sphere are a series of circles, far apart near the center (top) and near together…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a hammer.

Hammer

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a hammer.

Hammer

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a hammer.

Hammer

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a hammer.

Hammer

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

A hand holding a square pyramid, suspended by a string.

Hand and pyramid

A hand holding a square pyramid, suspended by a string.

"Hexagonal-rhombohedral. Crystals usually thick to thin tabular. Basal planes prominent, often showing triangular markings." — Ford, 1912

Hematite

"Hexagonal-rhombohedral. Crystals usually thick to thin tabular. Basal planes prominent, often showing…

"Hexagonal-rhombohedral. Crystals usually thick to thin tabular. Basal planes prominent, often showing triangular markings." — Ford, 1912

Hematite

"Hexagonal-rhombohedral. Crystals usually thick to thin tabular. Basal planes prominent, often showing…

"Hexagonal-rhombohedral. Crystals usually thick to thin tabular. Basal planes prominent, often showing triangular markings. Edges of plates sometimes beveled with rhombohedral and pyramidal forms." — Ford, 1912

Hematite

"Hexagonal-rhombohedral. Crystals usually thick to thin tabular. Basal planes prominent, often showing…

"A polygon of...seven sides [is] a heptagon." —Hallock 1905

Heptagon

"A polygon of...seven sides [is] a heptagon." —Hallock 1905

An illustration showing how to construct a heptagon, or septagon. "The appotem a in a hexagon is the length of the side of the heptagon. Set off AB equal to the radius of the circle; draw a from the center C at right angles to AB; then a is the required side of the heptagon."

Construction Of A Heptagon

An illustration showing how to construct a heptagon, or septagon. "The appotem a in a hexagon is the…

"A polygon of...six sides a [is] a hexagon." —Hallock 1905

Hexagon

"A polygon of...six sides a [is] a hexagon." —Hallock 1905

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a hexagon.

Hexagon

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a hexagon.

Hexagon

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a hexagon.

Hexagon

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a hexagon.

Hexagon

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

An illustration showing how to construct a hexagon in a given circle. "The radius of the circle is equal to the side of the hexagon."

Construction Of A Hexagon In A Circle

An illustration showing how to construct a hexagon in a given circle. "The radius of the circle is equal…

Illustration of a design made by 7 hexagons with adjacent sides.

Design Consisting of 7 Hexagons

Illustration of a design made by 7 hexagons with adjacent sides.

Illustration of a large hexagon with an interior design made by 7 hexagons with adjacent sides.

Large Hexagon Design Consisting of 7 Hexagons

Illustration of a large hexagon with an interior design made by 7 hexagons with adjacent sides.

A fourty-eight sided geometric shape.

Hexakisoctahedron

A fourty-eight sided geometric shape.

"The faces of the hexakistetrahedron correspond to one-half the faces of the hexoctahedron." — Ford, 1912

Hexakistetrahedron

"The faces of the hexakistetrahedron correspond to one-half the faces of the hexoctahedron." —…

"The hexoctahedron is a form composed of forty-eight triangular faces, each of which cuts differently on all three crystallographic axes. There are several hexoctahedrons, which have varying ratios of intersection with the axes." — Ford, 1912

Hexoctahedron

"The hexoctahedron is a form composed of forty-eight triangular faces, each of which cuts differently…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a hogfish.

Hogfish

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a hogfish.

Hogfish

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a hogfish.

Hogfish

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts a hogfish.

Hogfish

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Illustration of a horizontal line segment. A line segment is a section of a line with definite endpoints.

Segment, Horizontal Line

Illustration of a horizontal line segment. A line segment is a section of a line with definite endpoints.

Illustration of a horizontal line segment. A line segment is a section of a line with definite endpoints.

Segment, Horizontal Line

Illustration of a horizontal line segment. A line segment is a section of a line with definite endpoints.

Illustration of a horizontal line segment. A line segment is a section of a line with definite endpoints. The endpoints are marked A and B.

Segment, Horizontal Line

Illustration of a horizontal line segment. A line segment is a section of a line with definite endpoints.…

An illustration showing how to construct a hyperbola by plotting. "Having given the transverse axis BC, vertexes Aa, and foci ff'. Set off any desired number of parts on the axis below the focus, and number them 1,2,3,4,,5,etc. Take the distance a1 as radius, and, with f' as center, strike the cross 1 with f'1=a1. With the distance A1, and the focus f as center, strike the cross 1 with the radius F1=A1, and the cross 1 is a point in the hyperbola."

Construction Of A Hyperbola

An illustration showing how to construct a hyperbola by plotting. "Having given the transverse axis…

An illustration showing how to construct a hyperbola by a pencil and a string. "Having given the transverse axis BC, foci f' and f, and the vertexes A and a. Take a rule and fix it to a string at e; fix the other end of the string at the focus f. The length of the string should be such that when the rule R is in the position f'C, the loop of the string should reach to A; then move the rule on the focus f', and a pencil at P, stretching string, will trace the hyperbola."

Construction Of A Hyperbola

An illustration showing how to construct a hyperbola by a pencil and a string. "Having given the transverse…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts an ice skater.

Ice Skater

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts an ice skater.

Ice Skater

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts an ice skater.

Ice Skater

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures consisting of triangles, squares, and parallelograms are used to construct the given shape. This tangram depicts an ice skater.

Ice Skater

Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. Seven figures…

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the lengths of lines a and b are the same. You would need to measure to be sure of your conclusion.

Optical Illusions

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the…

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the lengths of lines a and b are the same. You would need to measure to be sure of your conclusion.

Optical Illusions

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the…

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the lengths of lines a and b are the same. You would need to measure to be sure of your conclusion.

Optical Illusions

Illustration showing optical illusions. It is not always possible to trust the eye to be sure if the…