**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

(

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

http://study.com/academy/lesson/radical-expression-definition-examples-quiz.html

Jackson, C. (n.d.). What is a Radical Function?- Definition, Examples and Graphs. Retrieved August

10, 2016 from

http://study.com/academy/lesson/what-is-a-radical-function-equations-and-graphs.html

Radical Functions. (n.d.). Retrieved August 12, 4016 from

https://www.wyzant.com/resources/lessons/math/precalculus/radical_functions

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