Illustration of a pyramid and with a regular polygon inscribed in and circumscribed about a cone. "If a pyramid whose base is a regular polygon is inscribed in or circumscribed about a circular cone, and if the number of sides of the base of the pyramid is indefinitely increased, the volume of the cone is the limit of the volume of the pyramid, and the lateral area of the cone is the limit of the lateral area of the pyramid."

Cone With Regular Polygon Inscribed and Circumscribed About

Illustration of a pyramid and with a regular polygon inscribed in and circumscribed about a cone. "If…

Illustration of a 12-point star (24-sided polygon) inscribed in a regular dodecagon. This can also be described as a regular dodecagon circumscribed about a 12-point star (24-sided polygon).

12-Point Star Inscribed In A Dodecagon

Illustration of a 12-point star (24-sided polygon) inscribed in a regular dodecagon. This can also be…

Illustration of an equilateral triangle that shows both the centroid (where the medians of the sides meet) and the incenter (where the angle bisectors meet).

Centers of Equilateral Triangle

Illustration of an equilateral triangle that shows both the centroid (where the medians of the sides…

Illustration of beginning of construction of a regular heptagon in a circle.

Construction of Regular Heptagon in a Circle

Illustration of beginning of construction of a regular heptagon in a circle.

Illustration of a regular nonagon. A nonagon is a closed geometric figure with 9 sides. A regular nonagon has 9 equal sides and 9 equal angles.

Regular Nonagon

Illustration of a regular nonagon. A nonagon is a closed geometric figure with 9 sides. A regular nonagon…

Illustration used to construct a regular pentagon on a given line.

Construction of Regular Pentagon

Illustration used to construct a regular pentagon on a given line.

Illustration of a decagonal prism with regular decagons for bases and rectangular faces. The hidden edges are shown.

Decagonal Prism

Illustration of a decagonal prism with regular decagons for bases and rectangular faces. The hidden…

Illustration of a right decagonal prism with regular decagons for bases and rectangular faces.

Decagonal Prism

Illustration of a right decagonal prism with regular decagons for bases and rectangular faces.

Illustration of a right decagonal prism with regular decagons for bases and rectangular faces. The prism is resting on its side, one of the faces.

Decagonal Prism Resting On Its Side

Illustration of a right decagonal prism with regular decagons for bases and rectangular faces. The prism…

Illustration of a swimming pool and water hose that is in the shape of a hollow regular decagonal prism with regular decagons on the ends/bases and square faces.

Swimming Pool Shaped Like A Decagonal Prism

Illustration of a swimming pool and water hose that is in the shape of a hollow regular decagonal prism…

Illustration of a hollow right heptagonal/septagonal prism with regular heptagons/septagons for bases and rectangular faces.

Heptagonal/Septagonal Prism

Illustration of a hollow right heptagonal/septagonal prism with regular heptagons/septagons for bases…

Illustration of a regular right heptagonal/septagonal prism with regular heptagons/septagons for bases and square faces. The hidden edges are shown.

Heptagonal/Septagonal Prism

Illustration of a regular right heptagonal/septagonal prism with regular heptagons/septagons for bases…

Illustration of a right heptagonal/septagonal prism with regular heptagons/septagons for bases and rectangular faces. The hidden edges are shown.

Heptagonal/Septagonal Prism

Illustration of a right heptagonal/septagonal prism with regular heptagons/septagons for bases and rectangular…

Illustration of a right heptagonal/septagonal prism with regular heptagons/septagons for bases and rectangular faces.

Heptagonal/Septagonal Prism

Illustration of a right heptagonal/septagonal prism with regular heptagons/septagons for bases and rectangular…

Illustration of a right nonagonal prism with regular nonagons for bases and rectangular faces. The hidden edges are shown.

Nonagonal Prism

Illustration of a right nonagonal prism with regular nonagons for bases and rectangular faces. The hidden…

Illustration of a right nonagonal prism with regular nonagons for bases and rectangular faces. The height of the prism is greater than the length of a side of the nonagon. The hidden edges are shown.

Nonagonal Prism

Illustration of a right nonagonal prism with regular nonagons for bases and rectangular faces. The height…

Illustration of a right octagonal prism with regular octagons for bases and rectangular faces. The hidden edges are shown.

Octagonal Prism

Illustration of a right octagonal prism with regular octagons for bases and rectangular faces. The hidden…

Illustration of a right octagonal prism with regular octagons for bases and rectangular faces. The height of the prism is greater than the length of a side of the octagon. The hidden edges are shown.

Octagonal Prism

Illustration of a right octagonal prism with regular octagons for bases and rectangular faces. The height…

Illustration of a right pentagonal prism with regular pentagons for bases and rectangular faces. The height of the prism is greater than the length of a side of the pentagon. The hidden edges are shown.

Pentagonal Prism

Illustration of a right pentagonal prism with regular pentagons for bases and rectangular faces. The…

Illustration of a right pentagonal prism with regular pentagons for bases and rectangular faces. The height of the prism is slightly greater than the length of a side of the pentagon. The hidden edges are shown.

Pentagonal Prism

Illustration of a right pentagonal prism with regular pentagons for bases and rectangular faces. The…

Illustration of a right pentagonal prism with regular pentagons for bases and rectangular faces. The height of the prism is slightly greater than the length of a side of the pentagon.

Pentagonal Prism

Illustration of a right pentagonal prism with regular pentagons for bases and rectangular faces. The…

Illustration of a right pentagonal prism with regular pentagons for bases and rectangular faces. The height of the prism is greater than the length of a side of the pentagon.

Pentagonal Prism

Illustration of a right pentagonal prism with regular pentagons for bases and rectangular faces. The…

Illustration of 2 similar right decagonal prisms. Both have regular decagons for bases and rectangular faces. The height of the prism and length of the side of the decagon on the smaller decagonal prism are one half that of the larger.

Similar Decagonal Prisms

Illustration of 2 similar right decagonal prisms. Both have regular decagons for bases and rectangular…

Illustration of 2 similar right heptagonal/septagonal prisms. Both have regular heptagons/septagons for bases and rectangular faces. The height of the prism and length of the side of the heptagon on the smaller heptagonal prism are one half that of the larger.

Similar Heptagonal/Septagonal Prisms

Illustration of 2 similar right heptagonal/septagonal prisms. Both have regular heptagons/septagons…

Illustration of 2 similar right hexagonal prisms. The height of the prism and length of the side of the hexagon on the smaller hexagonal prism are one half that of the larger.

Similar Hexagonal Prisms

Illustration of 2 similar right hexagonal prisms. The height of the prism and length of the side of…

Illustration of a right decagonal pyramid. The base is a decagon and the faces are isosceles triangles. The hidden edges are shown in this illustration.

Decagonal Pyramid

Illustration of a right decagonal pyramid. The base is a decagon and the faces are isosceles triangles.…

Illustration of a hollow right decagonal pyramid. The base is a decagon and the faces are isosceles triangles. The pyramid is inverted, meaning that the vertex is at the bottom and the base is on top.

Inverted Decagonal Pyramid

Illustration of a hollow right decagonal pyramid. The base is a decagon and the faces are isosceles…

Illustration of 2 right pentagonal pyramids with hidden edges shown. The pentagonal bases are congruent, but the height of the smaller pyramid is one half that of the larger.

2 Right Pentagonal Pyramids

Illustration of 2 right pentagonal pyramids with hidden edges shown. The pentagonal bases are congruent,…

Illustration of 2 right rectangular pyramids with hidden edges shown. The rectangular bases are congruent, but the height of the smaller pyramid is one half that of the larger.

2 Right Rectangular Pyramids

Illustration of 2 right rectangular pyramids with hidden edges shown. The rectangular bases are congruent,…

Illustration of 2 similar right pentagonal pyramids with hidden edges shown. The height of the pyramid and length of the side of the pentagon (base) on the smaller pentagonal pyramid are one half that of the larger.

Similar Pentagonal Pyramids

Illustration of 2 similar right pentagonal pyramids with hidden edges shown. The height of the pyramid…

"Regular reflection results from the incidence of radiant energy upon a polished surface. When a beam of light falls upon a mirror, the greater part of it is reflected in a definite direction as is illustrated, and forms an image of the object from which it came. A perfect mirror would be invisible." -Avery 1895

Regular Reflection

"Regular reflection results from the incidence of radiant energy upon a polished surface. When a beam…

"Science has succeeded in classifying the thousands of known crystals in six systems, to each of which belongs a number of forms having some property in common. In order to classify these different crystals, the existence of certain lines within the crystal, called axes, is assumed, around which the form can be symmetrically build up. These axes are assumed to intersect in the center of the crystal, and to pass through from one side to the other." — Hallock, 1905

Regular Octahedron

"Science has succeeded in classifying the thousands of known crystals in six systems, to each of which…

"Science has succeeded in classifying the thousands of known crystals in six systems, to each of which belongs a number of forms having some property in common. In order to classify these different crystals, the existence of certain lines within the crystal, called axes, is assumed, around which the form can be symmetrically build up. These axes are assumed to intersect in the center of the crystal, and to pass through from one side to the other." — Hallock, 1905

Regular Tetrahedron

"Science has succeeded in classifying the thousands of known crystals in six systems, to each of which…

"Science has succeeded in classifying the thousands of known crystals in six systems, to each of which belongs a number of forms having some property in common. In order to classify these different crystals, the existence of certain lines within the crystal, called axes, is assumed, around which the form can be symmetrically build up. These axes are assumed to intersect in the center of the crystal, and to pass through from one side to the other." — Hallock, 1905

Rhombic Dodecahedron

"Science has succeeded in classifying the thousands of known crystals in six systems, to each of which…