This analytical term paper is about the Solomon R Guggenheim Museum by Frank Lloyd Wright. The museum will be analyzed through Volume and Space Geometry. Meaning, the building will be disassembled for the purpose of discovering the two-dimensional (2D) and three-dimensional (3D) geometric forms and spaces of the building. The relationship between geometry and architecture is critical. In fact, architecture depends on geometry. "Geometry can exist without architecture but architecture cannot exist without geometry" (Blackwell 3). Just as geometry can be thought of in terms of planes and solids, so can architecture. Plane architecture is that which consists of 2D shapes, while solid architecture consists of 3D solids. As a starting point, the Guggenheim museum will be described in-depth. Next, the floor plan of the building will be disassembled into its 2D geometric forms and spaces. Then, the Monitor will be disassembled into its 3D geometric forms and spaces. Similarly, the connection between the Monitor and Main Gallery will be analyzed the same way. Finally, the Main Gallery will be disassembled into its 3D geometric forms and spaces. Also, the following will be provided for reference: 3D model diagrams, photographs, scaled floor plan, and scaled elevations.

II. Description

The Guggenheim Museum is an art gallery located on Fifth Avenue in New York. The museum was designed between 1943 and 1956 and completed in 1959. Frank Lloyd Wright, an architect and engineer, and Solomon R Guggenheim are the men that began the building process of the museum. It is built out of reinforced cast-in-place concrete and is painted in an off-white color. “In the Guggenheim Museum, perhaps Wright’s best known pubic building, the great skylit room (in this case an art gallery) is encircled by a continuous six-story ramp. The smaller three-story secondary mass, which Wright called a ‘Monitor’, contains the museum’s offices” (McCarter 211). In this urban site, Wright “shut out the surrounding environment altogether and replaced it with a beautiful inner space that was wholly artificial” (Riley 19). The building holds the Main Gallery, the Justin K Thannhauser galleries, the Monitor, an underground auditorium, an exhibition wing, a restaurant, and a bookstore. There are two vertical masses in the building. The large mass is the Main Gallery and the other is the Monitor. The characteristics of Modernism are illustrated in the Guggenheim Museum. First, the intersection of the masses is clear and readable from all points of view. Also, the surfaces of the masses do not have any ornamentation. Then, the representation of the surfaces and edges are continuous throughout the interior and exterior of the museum. From Perugi, the Guggenheim Museum is described: 

From viewing the floor plan, the geometric forms found in the Guggenheim Museum are circles and a square, an equilateral triangle and convex forms, and rectangles. A circle is defined as a “plane curve such that each point on the curve is the same distance from a fixed point. This point is called the center of the circle. The circle belongs to the class of curves known as conic sections because a circle can be described as the intersection of a right circular cone with a plane that is perpendicular to the axis of the cone” (Encarta). The Main Gallery and Monitor, the interior floors and corners of the curbs, the entrance and elevator shaft, the ramp and skylight are all in the form of a circle. Also, there are three concentric circles located in the main gallery. Concentric circles are circles having different perimeters but having the same center. The outermost circle is a full circle, while the inner two are part of a circle. The radius of the innermost circle is approximately half that of the outermost. The circle provides the most effective perimeter of all of the shapes because it provides the same area coverage as other shapes with the least perimeter (circumference). This is important because the greater the perimeter, the greater the cost of the building, so a circular building is the most economical. Next, a large room located in the southeast corner of the Main Gallery is a square. A square is an equal four-sided polygon. Its area is equal to bh, where b is the base and h is the height. A stairway shaft in the Main Gallery is an equilateral triangle. An equilateral triangle is an equal three-sided polygon. Its area is equal to (½)bh, where b is the base and h is the height. On the other hand, the stairway shaft in the Monitor is in a convex form. Also, the four columns located outside between the Monitor and Main Gallery are convex shaped. Finally, the city lot grid is a rectangle form. In addition, there are shafts and rooms in the Main Gallery that are rectangles. A rectangle is a parallelogram with four right angles. On the floor plan, the most noticeable geometric form is the circle.

The geometric forms seen in the small vertical mass, which Wright called the Monitor, are the circular cylinder, circular plate, and convex shaped cylinder. A circular cylinder consists of “two circular bases of equal area that are in parallel planes, and are connected by a lateral surface that intersects the boundaries of the bases” (Encarta). The volume of a cylinder is pr²h, where r is the radius of the circle and h is the height of the cylinder. The circular cylinder is cut into three layers. In particular, the top two layers are all glass windows. On top of the circular cylinder is a circular plate that has a larger radius than the cylinder. Also, on top of the circular plate is a hexagonal dome, which has the same center or origin as the circular cylinder. By definition, a hexagon is an equal six-sided polygon. A dome is similar to a sphere cut in half. Then, from the radius of the circular cylinder, a corner of a convex shaped cylinder is connected. A convex curves outward; it has a thick center and thinner edges. The convex cylinder is the highest geometric solid of the Monitor. The Monitor has the most clear geometric forms in the Guggenheim Museum.

The geometric forms seen in the connection between the Monitor and the Main Gallery are horizontal rectangular parallelopipeds, semicircular solids, and rectangular and square plates. A parallelopiped is “a prism that has parallelograms as the bases and a rectangular parallelopiped, or box, is one in which all six faces (four lateral faces and two bases) are rectangles” (Encarta). The volume of a box is the length multiplied by the width and height. There are four horizontal layers, which the lowest to the highest is 1 to 4. The first layer, which has the largest height, is in a “L” shape form when looking from above. It cuts through the circular cylinder of the Monitor and the inverted ziggurat of the Main Gallery. This layer combines two rectangular boxes and two semicircular forms. One semicircular solid radials around the circular cylinder of the Monitor. The diameter of this semicircular solid, which is also the width of the attached box, is larger than the diameter of the circular cylinder of the Monitor. The other semicircular solid protrudes out in the west direction on the southwest corner of the building. The edge of this semicircle and the end of the attached rectangular box are the same. The volume of a semicircular solid is equal to (½)pr²h. The other rectangular box cuts through the inverted ziggurat, which is seen protruding out on the southeast corner. The second layer, which has the same height as the middle layer of the circular cylinder of the Monitor, is a rectangular plate connecting the Monitor to the Main Gallery. One side of the plate connects to the edge of the circular cylinder. The width of the plate is smaller than the diameter of the circular cylinder. The third layer, which cuts between the middle and top layer of the circular cylinder, combines a square and rectangular plate. The centers of the square plate and circular cylinder of the Monitor are the same. The width of the rectangular plate is smaller than the square plate. Finally, the fourth layer, which has the same height as the top layer of the circular cylinder of the Monitor, is a rectangular box. Like the third layer, one side connects to the edge of the circular cylinder. The other side connects to the south corner of the convex cylinder. Altogether, these four layers create a sense of horizontality.

The geometric forms seen in the Main Gallery are a partial upside-down circular cone and spherical dome, an equilateral triangular cylinder and two square cylinders, a circular plate, three vertical plates, and boxes. The partial upside-down circular cone, or inverted ziggurat, is the most eye-catching geometric form in the Guggenheim Museum. A circular cone is a “surface generated by a straight line that moves along a closed curve while always passing through a fixed point. The straight line is called the generatrix, the fixed point is called the vertex, and the closed curve is called the directrix. If the generatrix is of infinite length, it generates two conical surfaces on opposite sides of the vertex” (Encarta). The volume of a circular cone is (1/3)pr²h. There are three circular grooves etched into the surface of the inverted ziggurat. These grooves have windows that provide light to the interior. The height of the inverted ziggurat decreases clockwise because there is a continuous spiraling ramp inside the circular cone. A spiral is a curve on a plane that continuously winds around a fixed point at a decreasing or increasing distance. Wright’s ziggurat is a “three-dimensional translation of the spiral and thus a primordial organic form” (Riley 50). The spiraling of the ramp is cantilevered by “eleven evenly spaced radial walls to carry the giant helicoid, the twelfth support being provided by the elevator and utility core. Adding to the spatial complexity of the work, Wright’s spiral ramp expanded in width as it rose both inward and outward, aligning its curved sides with two virtual cones with centers 270 feet above and 180 feet below the structure” (Riley 75). The spherical dome is centrally located on top of the inverted ziggurat. A sphere is a “surface of which every point is the same distance from a fixed point known as the center” (Encarta). The volume of a sphere is (4/3)pr³, where r is the radius of the sphere. The peak of the spherical dome is the highest point of the Guggenheim Museum. The equilateral triangular cylinder and the two square cylinders have the same height. The volume of an equilateral triangular and square cylinder is the area of its base times the height of the cylinder. The triangular cylinder connects to the edge of the inverted ziggurat, while the two square cylinders cut into the ziggurat. The circular plate is the base of the inverted ziggurat. It has the same height as the bottom layer of the circular cylinder of the Monitor. The vertical plates connect the two square cylinders to the equilateral triangular cylinder. There are two box forms in the Main Gallery. One is located on the southeast corner of the building. It cuts into the side of the inverted ziggurat. Using the radius of a circle round off its two vertical corners, which protrude out of the building. The Main Gallery has the most complex geometric form to describe, which is the inverted ziggurat.

IV. Conclusion

In conclusion, it should be evident that geometry is important in architecture. Wright’s challenge was to design an art museum with a lot of gallery space for a small rectangular city lot. He beat that challenge through the use of the circle and the spiral. The Guggenheim Museum is dominated by the use of circular forms and the spiral, which was Frank Lloyd Wright’s main interest in his last years. The Main Gallery is where the circle and the spiral are combined to form a seashell.

V. Bibliography

• Blackwell, William, A.I.A. Geometry in Architecture. Berkeley, CA: Key Curriculum Press, 1984.
• Doremus, Thomas. Frank Lloyd Wright and Le Corbusier: The Great Dialogue. New York: Van Nostrand Reinhold Company, 1985.
• McCarter, Robert. Frank Lloyd Wright: A Primer on Architectural Principles. New York: Princeton Architectural Press, 1991.
• Perugi, Liberto. The Solomon R. Guggenheim Museum: New York: Frank Lloyd Wright Architect. New York: The Solomon R. Guggenheim Foundation, 1975.
• Riley, Terence. Frank Lloyd Wright: Architect. New York: The Museum of Modern Art, 1994.
• Microsoft Encarta Encyclopedia 99. 1993-1998 Microsoft Corporation. All rights reserved.

Software: Design Work Shop Lite V1.8, 1998, Artifice
Note: Guggenheim 3D model downloaded here
Websites: Reference 1, Images 1 + 2