Geometry in Art and Architecture

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2. This stained glass window in the Holocaust Museum in Washington, DC demonstrates the use of triangles forming a hexagonal shape. Manipulation of lines structures through opposing midpoints creates a central vertex figure.This stained glass window in the Holocaust Museum in Washington, DC demonstrates the use of triangles forming a hexagonal shape. Manipulation of lines structures through opposing midpoints creates a central vertex figure.

3. In addition to housing locomotives, railroad depots were designed to handle passengers, freight, baggage, and mail. The Roundhouse at the Baltimore and Ohio Museum is the largest circular building in the world. Located in downtown Baltimore,the B and O railroad museum is the largest circular building in the world. With 22 sides and a 136 foot high ceiling, the roundhouse is geometric wonder.In addition to housing locomotives, railroad depots were designed to handle passengers, freight, baggage, and mail. The Roundhouse at the Baltimore and Ohio Museum is the largest circular building in the world. Located in downtown Baltimore,the B and O railroad museum is the largest circular building in the world. With 22 sides and a 136 foot high ceiling, the roundhouse is geometric wonder.

4. Arched windows and entranceways are use in railroad stations to allow incoming trains to enter and light for passengers and shoppers.Arched windows and entranceways are use in railroad stations to allow incoming trains to enter and light for passengers and shoppers.

5. The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa (Italy), the city with the famous Leaning Tower, about 1175 AD. Pisa was an important commercial town in its day and had links with many Mediterranean ports. Leonardo's father (Guglielmo Bonaccio) was a kind of customs officer in the North African town of Bugia now called Bougie where wax candles were exported to France. They are still called "bougies" in French, but the town is a ruin today says D E Smith (see below). So Leonardo grew up with a North African education under the Moors and later travelled extensively around the Mediterranean coast. He would have met with many merchants and learned of their systems of doing arithmetic. He soon realised the many advantages of the "Hindu-Arabic" system over all the others. D E Smith points out that another famous Italian - St Francis of Assisi (a nearby Italian town) - was also alive at the same time as Fibonacci: St Francis was born about 1182 (after Fibonacci's around 1175) and died in 1226 (before Fibonacci's death commonly assumed to be around 1250). He called himself Fibonacci [pronounced fib-on-arch-ee or fee-bur-narch-ee] short for filius Bonacci which means son of Bonacci. Since Fibonacci in Latin is "filius Bonacci" and means "the son of Bonacci", two early writers on Fibonacci (Boncompagni and Milanesi) regard Bonacci as the family name so that Fib-Bonacci is like the English names of Robin-son or John-son. Fibonacci himself wrote both "Bonacci" and "Bonaccii" as well as "Bonacij"! Others think Bonacci may be a kind of nick-name meaning "lucky son" (literally, "son of good fortune"). He is perhaps more correctly called Leonardo of Pisa or, using a latinisation of his name, Leonardo Pisano. Occasionally he also wrote Leonardo Bigollo since, in Tuscany, bigollo means a traveller. The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa (Italy), the city with the famous Leaning Tower, about 1175 AD. Pisa was an important commercial town in its day and had links with many Mediterranean ports. Leonardo's father (Guglielmo Bonaccio) was a kind of customs officer in the North African town of Bugia now called Bougie where wax candles were exported to France. They are still called "bougies" in French, but the town is a ruin today says D E Smith (see below). So Leonardo grew up with a North African education under the Moors and later travelled extensively around the Mediterranean coast. He would have met with many merchants and learned of their systems of doing arithmetic. He soon realised the many advantages of the "Hindu-Arabic" system over all the others. D E Smith points out that another famous Italian - St Francis of Assisi (a nearby Italian town) - was also alive at the same time as Fibonacci: St Francis was born about 1182 (after Fibonacci's around 1175) and died in 1226 (before Fibonacci's death commonly assumed to be around 1250). He called himself Fibonacci [pronounced fib-on-arch-ee or fee-bur-narch-ee] short for filius Bonacci which means son of Bonacci. Since Fibonacci in Latin is "filius Bonacci" and means "the son of Bonacci", two early writers on Fibonacci (Boncompagni and Milanesi) regard Bonacci as the family name so that Fib-Bonacci is like the English names of Robin-son or John-son. Fibonacci himself wrote both "Bonacci" and "Bonaccii" as well as "Bonacij"! Others think Bonacci may be a kind of nick-name meaning "lucky son" (literally, "son of good fortune"). He is perhaps more correctly called Leonardo of Pisa or, using a latinisation of his name, Leonardo Pisano. Occasionally he also wrote Leonardo Bigollo since, in Tuscany, bigollo means a traveller.

6. To the left is the famous �Mona Lisa�. Try drawing a rectangle around her face. Are the measurements in the golden proportion. To the left is the famous �Mona Lisa�. Try drawing a rectangle around her face. Are the measurements in the golden proportion.

7. The sriyantra ('great object') The diagram consists of nine interwoven isosceles triangles four point upwards, representing Sakti, the primordial female essence of dynamic energy, and five point downwards, representing Siva, the primordial male essence of static wisdom The triangles are ananged in such a way that they produce 43 subsidiary triangles, at the centre of the smallest of which there is a big dot (known as the bindu). These smaller triangles are supposed to form the abodes of different gods, whose names are sometimes entered in their respective places. In common with many depictions of the sriyantra, the one shown here has outer rings consisting of an eight-petalled lotus, enclosed by a sixteen petalled lotus, girdled in turn by three circles, all enclosed in a square with four doors, one on each side. The square represents the boundaries within which the deities reside, protected from the chaos and disorder of the outside world. The sriyantra ('great object') The diagram consists of nine interwoven isosceles triangles four point upwards, representing Sakti, the primordial female essence of dynamic energy, and five point downwards, representing Siva, the primordial male essence of static wisdom The triangles are ananged in such a way that they produce 43 subsidiary triangles, at the centre of the smallest of which there is a big dot (known as the bindu). These smaller triangles are supposed to form the abodes of different gods, whose names are sometimes entered in their respective places. In common with many depictions of the sriyantra, the one shown here has outer rings consisting of an eight-petalled lotus, enclosed by a sixteen petalled lotus, girdled in turn by three circles, all enclosed in a square with four doors, one on each side. The square represents the boundaries within which the deities reside, protected from the chaos and disorder of the outside world.

8. There are a great number of designs and patterns in art that have a firm footing in mathematics, especially geometry. Crafts like quilting, rug hooking, and cross stitching make good use of many geometrical concepts. By using a straight edge, a compass, a protractor and a sharp pencil, your students can create some very imaginative designs. There are a great number of designs and patterns in art that have a firm footing in mathematics, especially geometry. Crafts like quilting, rug hooking, and cross stitching make good use of many geometrical concepts. By using a straight edge, a compass, a protractor and a sharp pencil, your students can create some very imaginative designs.

9. �������� During the last decade of his life (1965-1975), Johnson devoted his time to creating abstract geometrical paintings, all of them based on mathematical theorems. According to his article "On the Mathematics of Geometry in My Abstract Paintings" (1972), Johnson began this work in 1961 "upon belatedly discovering aesthetic values in the Pythagorean right triangle and Euclidian geometry" (97). In all, he painted as many as 100 canvases, at least 60 of which are held by the Smithsonian Institution's National Museum of American History, Division of Information Technology and Society. Of the remaining paintings, some are privately held and others have been lost. �������� �������� During the last decade of his life (1965-1975), Johnson devoted his time to creating abstract geometrical paintings, all of them based on mathematical theorems. According to his article "On the Mathematics of Geometry in My Abstract Paintings" (1972), Johnson began this work in 1961 "upon belatedly discovering aesthetic values in the Pythagorean right triangle and Euclidian geometry" (97). In all, he painted as many as 100 canvases, at least 60 of which are held by the Smithsonian Institution's National Museum of American History, Division of Information Technology and Society. Of the remaining paintings, some are privately held and others have been lost. ��������

10. Escher was a great master of tessellation (the regular division of the plane, or tiling). He created symmetrical designs and planar tesselations, which he described as congruent, convex polygons joined together." Escher was a great master of tessellation (the regular division of the plane, or tiling). He created symmetrical designs and planar tesselations, which he described as congruent, convex polygons joined together."

11. Dutch artist, MC Escher,1893-1972 Major themes in Escher's work are contrast, duality, transformation, infinity and spatial paradoxes. He uses symmetry to order this world of duality and paradox. In the slide above Escher explores the duality of order vs. chaos. We shall see how this idea influences his work, both formally and psychologically. TilingEscher was a great master of tessellation (the regular division of the plane, or tiling). He created symmetrical designs and planar tesselations, which he described as congruent, convex polygons joined together." ImagesEscher always preferred to use only animate images in his tiled patterns. In 1922 Escher visited the Alhambra palace and saw the wall tilings of the Moors. He was excited to find other artists who had been captivated by tilings, but also made this revealing comment: "What a pity their religion forbade them to make graven images." Escher's notebooks soon became full of repeating patterns inspired by the Moors. Imagery gave his patterns a different psychological character from the serene designs of Islam. While, Escher's work includes representation, it is still involved with the language of visual symmetry and order. Symmetry is integral to the medium of printmaking and graphic arts. The impression of a woodblock is a reflection or mirror image of the design carved into the block. Multiplicity and repetition are functions of printing as well. Thus, Escher chose a medium that naturally expressed two motions of symmetry: reflection and translation. These elements of symmetry also showed Escher's strong love of order. The technical difficulty of woodcutting suited Escher's fastidious nature too. Escher was also fascinated by the concept of infinity, which led him into explorations of space beyond the two dimensional plane. He carved the surface of this six inch ball with twelve identical fishes to show that a "fragmentary" plane could be filled endlessly. "When you turn this ball in your hands, fish after fish appears in endless succession. Though their number is restricted, they symbolize the idea of boundlessness in a manner that is not obtainable." Dutch artist, MC Escher,1893-1972 Major themes in Escher's work are contrast, duality, transformation, infinity and spatial paradoxes. He uses symmetry to order this world of duality and paradox. In the slide above Escher explores the duality of order vs. chaos. We shall see how this idea influences his work, both formally and psychologically. TilingEscher was a great master of tessellation (the regular division of the plane, or tiling). He created symmetrical designs and planar tesselations, which he described as congruent, convex polygons joined together." ImagesEscher always preferred to use only animate images in his tiled patterns. In 1922 Escher visited the Alhambra palace and saw the wall tilings of the Moors. He was excited to find other artists who had been captivated by tilings, but also made this revealing comment: "What a pity their religion forbade them to make graven images." Escher's notebooks soon became full of repeating patterns inspired by the Moors. Imagery gave his patterns a different psychological character from the serene designs of Islam. While, Escher's work includes representation, it is still involved with the language of visual symmetry and order. Symmetry is integral to the medium of printmaking and graphic arts. The impression of a woodblock is a reflection or mirror image of the design carved into the block. Multiplicity and repetition are functions of printing as well. Thus, Escher chose a medium that naturally expressed two motions of symmetry: reflection and translation. These elements of symmetry also showed Escher's strong love of order. The technical difficulty of woodcutting suited Escher's fastidious nature too. Escher was also fascinated by the concept of infinity, which led him into explorations of space beyond the two dimensional plane. He carved the surface of this six inch ball with twelve identical fishes to show that a "fragmentary" plane could be filled endlessly. "When you turn this ball in your hands, fish after fish appears in endless succession. Though their number is restricted, they symbolize the idea of boundlessness in a manner that is not obtainable."

12. Here we have two different churches. Notice how both have a very similar design. Many churches have various styles of steeples. However, the shape and structure of the steeples varies. The one on the left uses many triangular designs while the one on the right uses right angles, triangular and semicircular shapes. Please note the clock on the Trinity Church. This clock is in the shape of an octagon.Here we have two different churches. Notice how both have a very similar design. Many churches have various styles of steeples. However, the shape and structure of the steeples varies. The one on the left uses many triangular designs while the one on the right uses right angles, triangular and semicircular shapes. Please note the clock on the Trinity Church. This clock is in the shape of an octagon.

14. Fort Jefferson is located on Garden Key in the Dry Tortugas, about 68 miles from Key West. The Dry Tortugas are part of the National Park system and can only be accessed by boat or sea plane. Construction of the fort began in 1846, and it was almost complete at the time the Civil War started. It was originally erected to be the largest fort in the coastal defense system with some 450 guns and 1500 men, however, the fort was never attacked and its value to coastal defense was rendered almost useless by the invention of the rifled cannon. After the Civil War, the fort served as a prison with its most famous prisoner being Dr.Samuel A. Mudd. Dr. Mudd was the physician that set the broken leg of James Wilkes Booth, the assassin of President Abraham Lincoln. It was abandoned for a period beginning in 1874, but subsequently served as a navy base (Spanish-American War), a seaplane port (WW I) and observation post (W.W.II). Fort Jefferson is located on Garden Key in the Dry Tortugas, about 68 miles from Key West. The Dry Tortugas are part of the National Park system and can only be accessed by boat or sea plane. Construction of the fort began in 1846, and it was almost complete at the time the Civil War started. It was originally erected to be the largest fort in the coastal defense system with some 450 guns and 1500 men, however, the fort was never attacked and its value to coastal defense was rendered almost useless by the invention of the rifled cannon. After the Civil War, the fort served as a prison with its most famous prisoner being Dr.Samuel A. Mudd. Dr. Mudd was the physician that set the broken leg of James Wilkes Booth, the assassin of President Abraham Lincoln. It was abandoned for a period beginning in 1874, but subsequently served as a navy base (Spanish-American War), a seaplane port (WW I) and observation post (W.W.II).