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# Mathematical Functions Used in Architecture

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## Michelle Domanog

on 29 August 2016

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#### Transcript of Mathematical Functions Used in Architecture

Mathematical Functions Used in Architecture
design by Dóri Sirály for Prezi
Domanog, Michelle Jeraldin
Go, Francesca Mae
Lim, Shane Belle Mae
Samaniego, Erika Mae
Tatoy, Aubrey Christine

IS THE FIELD THAT DEALS WITH
designing buildings
open areas
communities
other artificial constructions
environments

usually
with some regard to aesthetic effect.

A culminating project by:
General Form: f(x)= ax^2 + bx + c
- where a, b, and c are numbers with a not equal to zero
- domain is f(x) is the set of x-values for which the function is defined
- range of a function y = f(x) is the set of values y takes for all values of x within the domain of f(x).

Polynomial Functions
ARCHITECTURE
Architects may use the tangent function to compute for the structure’s height.

The following devices help measure angles:
- Clinometers (older devices)
- Digital technology to provide more accurate readings

a. Trigonometric Functions
b. Polynomial Functions
c. Logarithmic Functions
d. Rational Functions
e. Exponential Functions
Functions to Present:

Trigonometric Function
Trigonometric Function
Trigonometric Function
Inverse of the function
h = (tan θ)(d) is a trigonometric function where the dependent quantity is h (height)
and the independent quantities are tan theta or d (distance).

This function is derived from the tangent function, tan θ = opposite / adjacent or
tan θ = height / distance.

Domain:
For the quantity d:
Case 1.1: (0, ∞) when the height is above ground level because when it is above
ground, it is a positive height
Case 1.2: (-∞, 0) when the building also contains floors below ground
Case 1.3: (-∞, 0) U (0, ∞) when the building contains floors both above and below
ground level

For the quantity θ:
Case 2.1: (-∞, ∞) except ±π/2, ±3π/2, ±5π/2, …, (or in degrees: ±90°, ±270°, ±450°, …)

Range:
(-∞, ∞)

h = (tan θ)(d), where θ = 60°

Note: θ can be equal to any value from 0 - 90 degrees,
depending on the angle measured by the device.

Graph may also change, depending on the value of θ.

Polynomial Function
To graph
f(x) = ax^2+ bx + c, which can be written in vertex form which is f(x) = a(x - h)^2 + k
where h = - b / 2a and k = f(h)

In order to determine the direction of the opening in the parabola, observe the sign in a. If a > 0, then it opens upward; if a < 0, then it opens downward.
Sydney Harbour Bridge
Sydney Harbour Bridge
The “Scale Factor” was derived by knowing that the real life span distance of the roadway
below the arch was 503 meters, and then comparing this with the number of pixels on the photo for this span.

Sydney Harbour Bridge
With the values above, it is assumed that
the vertex (h,k) is equivalent to (251.5,118)
Three other (x,y) points on the arch are:
(323,108) and (394.5,80) and (503,0)

Sydney Harbour Bridge
Determine the Dilation Factor value or “a” in the equation
f(x) = a(x - h)^2 + k
Given: Vertex (251.5,118) Domain: all real numbers
x intercept: (503,0) ; y intercept (0,0) Range: (-

, 118]
Find: a
Solution: f(x) = a(x - h)^2 + k Leading to the vertex form
y-k= a(x - h)^2
f(x) = -0.00188(x - 251.5)^2 + 108

0-118 = a(503 - 251.5)^2
-118=-62749.25a
a=0.001880501 -0.00188

f(x) = a(x - h)^2 + k
Logarithmic Functions
Exponential Functions
Rational Functions
A rational function is defined as the quotient of two polynomial functions.

f(x) = P(x) / Q(x)
Function of the center tower
Asymptotes:
HA at y= 0
VA at x=± 0.6325
Domain:
(-∞, -√10/5) U (-√10/5,√10/5) U (√10/5, ∞)
Restricted domain:
-0.799 <= x <= -0.655
0.655 <= x <= 0.799
No zeros
Aside from using functions to design, architects also use functions to estimate the deterioration of materials by using the Exponential Decay or Growth Formula:

f(x)= 1

2
5x -2
The function has no asymptotes and has a zero at (0,0).
A certain strain of bacteria grows in a certain of wood you're planning to use, doubles every 5 minutes. Assuming that you start with only one bacterium, how many bacteria could be present after 96 minutes?
Now that we have the value of
k
, we can find the number of bacteria in 96 minutes.
The domain is the set of all real numbers

The range of this function is from (0, ∞)

The function has no zeroes but has a horizontal asymptote of Q = 0
2 = 1
e
kt
2 =
e
k(5)
ln 2 = ln
e
(5)k
ln 2 =
5k
ln
e
ln 2 =
5k
5 5
k = .1368294361
Q = 1
e
(.136294361)(96)
Q = 602, 248.7625 bacteria
where
Q is the initial amount at time 0:
Q(0) = Q
k
is the rate of growth or decay in decimal,
and
e
is the base.
t
is the term over which the growth or decay occurs.
Q
(
t
)=
Q e

kt
0
, for t ≥ 0 ,
for t ≥ 0
Q
(
t
)=
Q e
kt
0
0
0
The independent quantity is
t
because it does not rely on any other variables.

On the other hand, Q(t) is the dependent quantity because there has to be a given value in order to solve for it.
Q
(
t
)=
Q e
kt
0
EXAMPLE
Q
(
t
)=
Q e
kt
0
Q
(
t
)=
Q e
kt
0
This function is an example of Exponential Function because the constant,
e
, is always raised to the variables
k
and
t
.
The inverse of this function is the Logarithmic Function. With the formula given, we can derive it as:
Trigonometric Function
ln =
k
(
t
)
Q
0
Q
Q
=
Q e
kt
0
ln
Q
= ln
Q +
ln
e
kt
0
ln Q – ln Q = ln e
0
kt
ln =
k
(
t
) ln e
Q
0
Q
ln =
k
(
t
)
Q
0
Q
Trigonometric Function
In a right triangle, there are actually six possible trigonometric ratios, or functions. A Greek letter (such as theta θ or phi φ) will be used to represent the angle.
Logarithmic functions are used in
Hardware Architecture
Hardware Architecture
is the identification of a system’s physical components and their interrelationships.

A
Parabolic Synthesis
is an implemented approximations of unary functions in hardware.
It is used to develop sub functions that will create a parabolic synthesis.
Unary functions
, like logarithmic functions are extensively used in computer graphics, digital signal processing, communication systems, robotics, astrophysics, fluid physics, etc.
Documentation
References
Lauriano, I. (2016, February 26). Trigonometry in architecture. Retrieved August 13, 2016, from
https://issuu.com/ivanlauriano/docs/trigonometry_in_architecture_655fdca3d68a33
https://hal.archives-ouvertes.fr/hal-01166872/document
Beddoe, J. (n.d.). Radical Expressions: Definitions and Examples. Retrieved August 10, 2016 from
Jackson, C. (n.d.). What is a Radical Function?- Definition, Examples and Graphs. Retrieved August
10, 2016 from
Radical Functions. (n.d.). Retrieved August 12, 4016 from
Passy. (2012, October 13). Sydney Harbour Bridge Mathematics. Retrieved August 14, 2016, from Passy's World of Mathematics:
http://passyworldofmathematics.com/sydney-harbour-bridge-mathematics/
Desmos Graphing Calculator
Graphs of Exponential Growth/Decay. (n.d.). Retrieved August 10, 2016, from
http://serc.carleton.edu/introgeo/teachingwdata/Graphsexponential.html
Inverse, Exponential, and Logarithmic Functions. (n.d.). Retrieved August 10, 2016, from
http://www.sparknotes.com/math/calcab/logs/section4.rhtml
https://mycourses.aalto.fi/pluginfile.php/132017/mod_resource/content/1/Architectural%20Geometry_lecture02_20150918_DrT.pdf
The function isn't a one-to one function
Original Function for Developing Sub-functions
The function f(x)=log (x+1) is used in developing sub functions to facilitate the hardware implementation by limiting the numerical range.

In this function, the independent variable is x and the dependent variable is (x+1)
domain: (-1,∞)
range: (-∞,∞)
asymptotes: none
Graph of the Function
Inverse of the Function
The inverse of the function f(x)=log (x+1) is 2 =1
2
2
x
The First Sub-Function
The first sub-function, s (x), is developed by dividing the original function, f(x), with x as an approximation.
As shown in this graph there are two possible results after dividing the original function with x, one where f(x)>1 and one where f(x)<1
S (x)=
log (x+1)
x
1
1
2
The Second and Third Function
f(x)= 1

5x - 2
2
Inverse of the function
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