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Enduring Urban Forms
Published in Lisa Heschong, Visual Delight in Architecture, 2021
The largest of all DC government buildings, the Pentagon, headquarters of the US Department of Defense, broke the mold of alphabet shapes when it was built in 1941. Instead, the massive five-sided building sports five pentagonal rings, one inside the other, each 50 feet wide and five stories high. Dozens of courtyards separate each ring of the pentagon. The Pentagon’s thousands upon thousands of office windows face each other squarely across these narrow courtyards, in military precision. Apparently, visual and acoustic privacy were not a concern of the military at that time.
Hazmat Team Spotlight
Published in Robert A. Burke, Hazmat Team Spotlight, 2020
The Pentagon, located in Arlington County, VA, is the headquarters of the U.S. Department of Defense and the world’s largest office building in terms of square footage 6.5 million square feet, of which 3.7 million is used for offices. The building has five floors above ground and two below ground. Approximately 23,000 military personnel are employed at the Pentagon, along with 3,000 non-defense support personnel.
Natural gas hydrate
Published in Jon Steinar Gudmundsson, Flow Assurance Solids in Oil and Gas Production, 2017
A water molecule is located at each of the corners of a hydrate dodecahedron in a tree-dimensional structure. The oxygen atom of water is larger than each of the hydrogen atoms. The molecule is highly polar and the hydrogen atoms a separated by an angle of 105 degrees. The angle of each of the corners of an ideal pentagon is 108 degrees.
Soap films and GeoGebra in the study of Fermat and Steiner points
Published in International Journal of Mathematical Education in Science and Technology, 2018
In preparation for computing the length of the pattern joining the vertices of the pentagon, students reviewed some properties of regular pentagon, namely, the ratio of the length of a diagonal to the length of the side (golden ratio), angles formed by sides and diagonals, and several similar triangles within the pentagon formed by the diagonals and sides. Students also computed the sum of distances from the Fermat point for the 72˚–72˚–36˚ triangle, as well as the angles formed by the soap film and the sides of such a triangle (Figure 21). Students and teams used different strategies to compute the lengths of the unknown segments, including similar triangles, and trigonometric relations. Often the strategies of the students were different from the strategy the instructor had envisioned. Teams shared their strategies, and sometimes they would adopt a strategy from the other team, such as using the law of sines (Figure 22).
Paper folding and trigonometric ratios
Published in International Journal of Mathematical Education in Science and Technology, 2019
Before we can find the trigonometric ratios of the following angles: 27°, and 63°, we need to understand a few properties of a regular pentagon. Let polygon ABCDE be a regular pentagon in Figure 2. Let us join point C to points A and E, and then choose a point F on AC such that FC = BC. Let G be the midpoint AE. Let the length of each side of pentagon ABCDE be 1. Since each angle of a regular pentagon is 108°, ∠BCA = ∠BAC = ∠DCE = ∠DEC = 36°. This implies ∠ACE = 108° – 2(36°) = 36°.
The golden ratio and regular hexagons*
Published in International Journal of Mathematical Education in Science and Technology, 2020
Another well-known occurrence of golden ratio takes place in the context of regular pentagons. In Figure 3, suppose polygon ABCDE be a regular pentagon with diagonals AC and CE. Since each angle of a regular pentagon is 108°, we can show ∠AEC = ∠EAC = 72°, and ∠ACE = 36°. Therefore, AE: EC = φ: 1.