Final answer:
In a newly designed building, windows can come in shapes of rectangles, circles, trapezoids, and right triangles, each with their unique calculations for areas and architectural implications. Gothic architecture, such as the rose window, illustrates the use of geometry in building design, and mathematical proportions influence the spatial organization within structures.
Step-by-step explanation:
Within the context of a newly designed building, windows can take on various shapes, each bringing a different architectural aesthetic and mathematical complexity. For example, windows in the shape of rectangles would be defined by their length and width and computed using basic geometry for area. Circular windows, like the rose windows mentioned, which are typical of Gothic architecture, may utilize plate tracery and have an area that can be calculated using the formula πr2 (pi times the radius squared). Trapezoids require a more complex calculation involving the lengths of both bases and the height, while right triangles can have their area found through ½ base times height. The mention of Gothic architecture elements like the rose window and lancet windows gives us a real-world reference to the application of these geometric shapes and forms in building design.
Furthermore, in this modern building, the general proportions follow a mathematical expression given as X = Y x 2 + 1. This expression could be reflected in the design of window shapes and their placements. The inclusion of ratios, such as the one guiding the spacing between columns, illustrates how mathematics is seamlessly integrated into architectural decisions, reinforcing the truth that math is everywhere.