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Fundamentals
Published in Michael Hann, The Grammar of Pattern, 2019
A diagonal in a square of sides equal to one unit can generate two 1:√2 right-angled triangles (Figure 2.3). The proportion 1:√2 is used commonly in the visual arts (Hiscock 2007, p. 201). A square set at 45° inside a larger square (alternatives shown in Figure 2.4) is known as the ad quadratum (Hiscock 2007, p. 187). This type of division creates a series of right-angled isosceles triangles with inner angles of 45°, 45° and 90°, with each triangle featuring a side proportion of 1:√2 (Hiscock 2007, p. 187). Two equal-size squares can create an eight-point star, by simply placing one square on top of the other (initially with sides and angles coinciding) and rotating it by 45° (Figure 2.5). Also, the overlapped area of the two squares creates a regular octagon; in recent research conducted by Wang (2017), this octagon shape was found to be a common construction in European medieval cathedral floor plans.
Areas of common shapes
Published in John Bird, Basic Engineering Mathematics, 2017
An octagon is an 8-sided polygon. If radii are drawn from the centre of the polygon to the vertices then 8 equal triangles are produced, as shown in Fig. 26.26. Area of one triangle=12×base×height=12×5×122=15cm2Areaofoctagon=8×15=120cm2 $$ \begin{aligned} \text{ Area} \text{ of} \text{ one} \text{ triangle}&= \frac{1}{2} \times \text{ base} \times \text{ height}\\&= \frac{1}{2} \times 5 \times \frac{12}{2} =15\,\mathrm cm ^{2}\\ \mathbf{Area of octagon}&= 8 \times 15 = \mathbf{120}\,\mathbf cm ^\mathbf{2} \end{aligned} $$
Expected distances and alternative design configurations for automated guided vehicle-based order picking systems
Published in International Journal of Production Research, 2022
Francisco J. Aldarondo, Yavuz A. Bozer
Appendix G presents an approximate analytic result for in •/CIRC/RAN/•. The approximation is based on deriving the exact solution to for an octagon-shaped FA with the same area as the circle. The octagon is constructed from a square (that envelops the circle) by removing the square's four corners as shown in Figure G1 in Appendix G. Each such corner consists of an isosceles triangle with sides , , and . For a circle with a radius of and area of , c is set equal to , resulting in an octagon of area . The Monte Carlo results for – based on 10,000 trips per replication and 10 replications – show that the approximation works well. In a FA of unit area, the mean (standard deviation) is equal to 0.813 (0.001) and 0.816 (0.001) for •/CIRC/RAN/• and respectively. This represents an error of less than 1% when replacing the circle with an octagon of the same area.
The Use of Geometrical Tracing, Module and Proportions in Design and Construction, from Antiquity to the 18th Century
Published in International Journal of Architectural Heritage, 2022
In the architecture of the Renaissance, the use of the octagon and the quadratura is of great importance, as it is proven by the works of Brunelleschi, Albert, Leonardo Da Vinci and Palladio. Andrea Palladio was influenced by Greek and Roman architecture, primarily Vitruvius (Palladio [1570] 1965). In the Villa Rotonda the rotating squares define the simple proportions of the plan and the elevation. These are greatly influenced by the theory of Alberti and the importance of Pythagorean intervals and numbers in general.