What are the greatest (and smartest) applications of mathematics in architecture? Architecture is inherently connected to Math in a fundamental way. My teacher asked me to prepare a presentation on Math and Architecture and the relationship they share. So, I've been gathering info from the Internet ( almost every site mentions the golden ratio ). 
So, I want to know about the most manipulative, smartest and most powerful applications of Math in architecture to introduce a "wow-factor" to my presentation. I'm not sure if this is the correct place to ask this question, but I'm at a loss where else to ask this. 
 A: One of them is the golden ratio $\frac{\sqrt{5}+1}{2}$.

Some twentieth-century artists and architects, including Le Corbusier and Dalí, have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts.

https://en.m.wikipedia.org/wiki/Golden_ratio
When walking into a room whose dimensions are the same proportion as the golden ratio, after watching the room, their eyes 'light up'. My opinion is that the brain recognises the proportion.
A: You might like to check out https://en.m.wikipedia.org/wiki/Parabolic_arch for example which explains the mathematical reasons for why the parabolic arch is used.
A: I think the plastic number and the Padovan sequence are of particular importance in architecture. The subject was covered by Padovan himself in this article:
R. Padovan , "Dom Hans van Der Laan and the Plastic Number," pp. 181–193 in Nexus IV: Architecture and Mathematics, Kim Williams and Jose Francisco Rodrigues, eds. Fucecchio (Florence): Kim Williams Books, 2002.
More generally, I found these books on Amazon: 
Architecture and Mathematics from Antiquity to the Future: Volume I: Antiquity to the 1500s 2015th Edition
Architecture and Mathematics from Antiquity to the Future: Volume II: The 1500s to the Future 2015th Edition
These look interesting, I'd like to see them myself. By the way, Padovan's article is also in Volume II.
A: I thought immediately about the Penrose tiling as the base of architectural form generation:

Indeed, there is a journal of Architecture and Mathematics, the "Nexus Network Journal", where I found an excerpt of an article regarding Penrose tiling: "Michael J. Ostwald - Aperiodic Tiling, Penrose Tiling and the Generation of Architectural Forms", which talks about "Penrose tiling in the context of architectural form generation".
For instance here is a nice article from ScienceNews regarding the Penrose tiling concept in ancient Islamic architecture.
A: From an artistic point of view, it is said that the golden ratio governs the dimensions of the Parthenon in Rome as is described here:
https://www.goldennumber.net/parthenon-phi-golden-ratio/
However maths is used in a fundamental way to simply measure and calculate throughout the process of building design - particularly with respect to materials estimation, design (i.e. trigonometry, geometry), and even in simple form, with regards to cost, time and materials estimation.
The most powerful applications of maths are probably within the CAD software which is used both for engineering calculations, and even more intensively for the rapid design and visualisation of buildings nowadays.
In order to render a 3d-vis of a modern building - especially an animated video - including all the materials, textures, and raytracing light and shadows it is not unusual to use a rendering farm with hundreds of CPU's, running for hours or days to produce the final output.  All of this computation is maths - trillions of calculations and you should be able to find some nice videos online for your presentation.  Of course the architect is insulated from all this maths by the software (s)he uses.
Maths has also been used design-wise for aesthetic effect through the years; as in the example of the Parthenon.  Also of particular mathematical interest are the tiling patterns used by the moors.  The ancient arabs were excellent mathematicians - the al gebra is an arabic word, and there are complex tesselations in use in many of their buildings which demonstrate their skill.
That maths is used throughout architecture should be of no surprise, since maths is in many ways, the study of form and as such it pervades most fields of life. But the built environment is so inherent to our lives and we place so many demands on our buildings and infrastructure, that it is inevitable we will want to accurately calculate, in order to achieve the performance we desire.
