It was recently brought to our attention that the site which was hosting a lot of the images and PDF documents for Passy’s World of Mathematics got hacked, and there has been some very unsavoury advertising and spam present on some of our lessons.
We are slowly repairing the site, and doing work to host the images on our own server space. This could easily take us the next 30 days to fix all 190 lessons on this site. We thank you for your patience.
Mathematics is just one of my passions. As well as teaching Mathematics, MultiMedia, and ICT as my “Day Job”, I also work as a Photographer on weekends.
Modern Digital Photography involves quite a bit of behind the scenes Mathematics.
This Mathematics includes Geometric Sequences, Polyhedrons, Fractions, Geometry, Circle Areas, Angles, Algebra Formulas, and Symmetry.
An understanding of this Mathematics can definitely make you a better Photographer, even if it just means you know the best Viewpoints, Angles, and Geometric Composition to use when taking pictures with a Mobile Phone.
I have found that understanding the Mathematics of a DSLR Camera really helps me do fast effective problem solving on Photoshoots.
Digital Cameras have preset auto modes which work well in about 80% of situations, but there are other times when a Camera needs to be put into full Manual Mode and some Mathematical Problem Solving invoked.
Here at Passy’s World we have produced a comprehensive series of articles on the Mathematics of Photography, including plenty of real life examples as well as selected Videos which can be viewed.
We recommend working through these articles in the order in which they appear in the Overview Below.
The eight Mathematics of Photography articles are as follows:
– Photo Composition Rules
– Digital Camera Variables
– The ISO Variable
– The Aperture Variable
– The Shutter Speed Variable
– The White Balance Variable
– Combining Variables for Correct Exposure
– Flash Photograpy and Diffusers
Contained below is an outline of each article followed by a clickable link to the full article.
It is important to work through the articles in the order they are given here.
Like great Works of Art, the elements in a Photograph need to be arranged in a manner which is interesting and pleasing to the eye.
In addition, Photography is a two dimensional medium. Therefore we need to make use of the elements which are present in the photo to create a sense of depth and three dimensions.
Photos also need a main focal point, as well as leading our eyes on a journey through the picture.
Having these things makes for great photos, which are pleasing to the eye, and we call this “Composition”.
There are a Number of Composition Rules involving Percentage, Geometry, Symmetry, a Grid of Thirds, and even the “Golden Ratio” and its spiral.
To find out about these rules and their Mathematics, click the link below:
Most people use their camera on “Auto” mode, and get plenty of good pictures.
However, there are many items on a camera which can be adjusted manually to get even better pictures!
In this “How To” article we introduce several of these camera settings, (known as “Variables”), and show how they can be used to create quality photos.
An overview of the three main Variables: “ISO Light Sensitivity”, “Aperture”, and “Shutter Speed” is provided.
This provides the preliminary background to more detailed articles on each of the Variables.
It is therefore vitally important to read this article before proceeding forward to the specific detailed articles.
To read the Introductory Article about Camera Variables, click the link below.
Digital Camera Variables and Settings
ISO sets a Camera’s sensitivity to light
Effectively it works like a Brightness Control on your camera, and you can turn it up higher if you are in a dark room, or in a shady forest.
If you are in Bright Sunlight, then you need to turn the ISO down to a low value or else your photo will come out all white and washed out with too much brightness.
Camera ISO is one of the three key Variables of Photography, the other two being Aperture and Shutter Speed.
Every photographer needs to understand ISO in order to get bright and clear pictures from their equipment.
ISO values increase in a “Geometric Progression” or a “Geometric Sequence” (Factor of Two, which results in Doubling).
This ISO sequence is: 100, 200, 400, 800, 1600, 3200, 6400 and etc. For every step up in this scale, the brightness of your picture doubles.
To get the full story on ISO and how to set it correctly on Digital Cameras for a variety of lighting conditions, click the link below.
Aperture is one of the key variables in Digital Photography. The other two are ISO Light Sensitivity and Shutter Speed.
An understanding of Aperture is critically important to isolate subjects in portraits and get full detail in Landscape pictures.
The Aperture Scale consists of a seemingly odd set of decimal and whole numbers, and getting your head around these values is not at all easy for Beginner Photographers.
In this article we examine Aperture in detail, and mathematically explain where the Aperture “f-numbers” come from.
Click the following link to read the full article.
Shutter Speed is mainly used for creating dramatic effects by either freezing action or blurring motion.
The Speed is expressed in a fraction of a second, and is how long we let light in through the camera lens to the sensor.
Think of the Shutter as like opening and closing a set of venetian blinds quickly to let a burst of light into a darkened room.
Shutter Speed forms a Geometric Progression of Fractions with a Common Ratio of 1/2.
Shutter Speed, (also called “Exposure Time”), becomes critically important in situations outside of the norm, such as the following:
1) Fast Moving Sports Action
2) Dimly lit indoor areas such as Bars and Clubs
3) Night Time Photography
4) Theatre and Musical Performances
5) Indoor Photos using Flash
6) Indoors Photos where we do not use Flash
7) Photos where we want motion blur for fast moving objects
8) Bird and Wildlife Photography
9) Photographing Young Children and Animals
In this article we will show you what Shutter Speed is, what the speed numbers mean, and how to set shutter speeds to values which should produce great photos.
Click the following link to read our article on Shutter Speed.
Different light sources produce light with slightly different colour tints, but our eyes do a great job correcting these variations.
Humans do not see a shift in colour as we move from a sunny garden into a shaded area, or go into a room and turn a light on.
Generally wherever we are, a piece of plain white paper always looks white.
However Digital Cameras DO detect light source differences, and sometimes create pictures with incorrect looking colours in them.
We need to know how to perform “white balance” adjustments, so that we can obtain pictures with realistic colours in them.
White Balance involves the Primary Colours of the Spectrum forming a Kelvin Temperature Scale from “cool” blue/violet through to warm “red” candle light.
What we have is a Mathematical Scale of color, and White Balance blends various colours together so that a white sheet of paper will look white, no matter what coloured lighting we are in.
You can read the full article on White Balance at the following link:
In this article we look at Combining Variables for Correct “Exposure”.
We tie together the three key Digital Camera Variables: ISO, Shutter Speed, and Aperture, and discuss how to set these up to get good exposures in Camera Manual Mode.
When these three variables are optimised and unified together, we get a nice clear picture with great contrast and highlights. We say that the photo is correctly “Exposed”.
You can read the full article on Getting Correct Exposure Using Key Variables at the following link:
Combining Variables for Correct Exposure
In this article we discuss Flash Photography and how to use Light Diffusers to make Flash Photos look so much better.
Diffusers come in all types of Geometric Shapes and Sizes and create softened light which is more like Daylight.
The Intensity of Light from a point source like an on camera Flash follows an Inverse Square Law with the Distance the Flash is from the subject.
This is very important to have an intuitive idea about Inverse Square Law when changing the Power on a Flash, or when we move further away from the subject being photographed.
The full article about Flash and Diffusers can be read at the following link:
Flash Photograpy and Diffusers
Working through this series of articles will definitely make you a better photographer.
Keep in mind that I spent a couple of years working with Digital Photography to reach this level of understanding to a point where I could write about it to share my knowledge with others.
So if you are a complete beginner and feel a bit overwhelmed, then that is okay, these things just take practice and time.
It is best to work through all of our eight articles a bit at a time, try things out, and then come back to the articles to pick up a bit more of the finer detail.
These articles can also be great to refer to when a student says something like: “I don’t need to know Mathematics, I’m going to be a Photographer!”.
Well there is a lot of Mathematics involved with Photography, Cameras, and Lighting; even if you are only the model being put into poses and surrounded by intense studio lights.
The Maths is helpful, but the main thing about being a Photographer is to get out there taking lots of pictures!
Life is short and there can never be too many photos of its little magic moments!
Enjoy,
Passy
Photos By Passy
My Photography website can be found at the following link:
If you would like to check out a Portfolio of my Photography Work, then check out the Photo Albums on Flickr at the following link:
Photos By Passy Portfolio Albums
Finally if you are interested in reading more “How To” articles about Photography, then visit the “How To” section on my website at this link:
Photos By Passy How To Articles
Image Copyright 2013 by Passy’s World of Mathematics
We have a great set of 12 color Posters covering the Exponent Laws that is available for free to all subsribers to Passy’s World.
If you are not already a subscriber, then click the following link to find out how to become an email subscriber.
(It is totally free to subscribe).
http://passyworldofmathematics.com/subscribe/
If you would like to see a preview of the full set of posters, then view the following SlideShare presentation.
[slideshare id=34150827&doc=exponentsrulespptslides-140430205536-phpapp02]
Image Copyright 2013 by Passy’s World of Mathematics
The posters are available as either a PDF or a PPT Presentation.
We suggest you then print them in color on standard A4 or Letter sized paper, and then use a Color Photocopy to enlarge them to Double Size or Larger.
If you are a subscriber you can get this Poster Set for free by emailing Passy’s World at the link shown below:
Related Items
Why Exponents and Indices are Important
Like Us on Facebook
Our Facebook page has many additional items which are not posted to this website.
These include items of mathematical interest, funny math pictures and cartoons, as well as occassional glimpses into the personal life of “Passy”.
Check it out at the following link:
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While you are there, LIKE the page so you can receive our FB updates to your Facebook News Feed.
Help Passy’s World Grow
Each day Passy’s World provides hundreds of people with mathematics lessons free of charge.
Help us to maintain this free service and keep it growing.
Donate any amount from $2 upwards through PayPal by clicking the PayPal image below. Thank you!
PayPal does accept Credit Cards, but you will have to supply an email address and password so that PayPal can create a PayPal account for you to process the transaction through. There will be no processing fee charged to you by this action, as PayPal deducts a fee from your donation before it reaches Passy’s World.
Enjoy,
Passy
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In this lesson we look at Exponential Growth of Populations.
Exponential growth involves increases starting off as reasonably small, and then dramatically increasing at a faster and faster rate.
The world’s accelerating population growth is a major concern in terms of how our planet can feed and provide fuel for the current 7.2 billion plus people who currently live in our world.
In addition to the explosion of humans on our own planet, Animals and Bacteria also increase exponentially.
The following video shows how Bacteria divide and multiply exponentially.
Human Population Growth
By the year 2000, there were around 10 times more people on Earth than there were just 300 years ago in 1700.
WORLD POPULATION
Year —– Population
1700 —– 600 000 000
1800 —– 900 000 000
1900 —- 1 500,000 000
2000 —- 6 000,000 000
2011 —- 7 000,000 000
On a graph, the increase looks like this:
Image Source: http://www.pfaf.org
Here is an excellent two and a half minute video which shows the history of the world’s population increase:
Increasing Life Expectancy
Image Source: http://elimfamilychurch-eastbourne.org.uk
The death rate has reduced, and people now live a lot longer and many more children are produced and live longer.
In 1960 the average age of death was 53 years old, but now it is around 75 years old.
The increase in health and life expectancy was historically unevenly spread throughout the world.
But over the last 200 years, Long Life and Good Health have generally increased throughout most of the world, and this has contributed to our exponentially huge population increase.
The following four minute Swedish video shows what has happened over the last 200 years.
Population Growth Equation
Image Copyright 2013 by Passy’s World of Mathematics
Population Growth follows Mathematics which involves exponents.
Exponential increases start off slow, but then sharply increase to a massive explosion in size, just like the power of a rocket engine igniting at take off.
An exponents formula, similar to the one used on compound growth for superannuation and interest bearing investments, can be used to estimate the Populations of Humans, Animals, and Bacteria.
Image Copyright 2013 by Passy’s World of Mathematics
This Equation involves the exponents of Rate x Time, and this is why Exponential patterns of increase in Populations occur.
Find out more about this equation at the following link:
Click here for Population Growth Mathematical Equations
There is a set of Population Growth fully worked example Maths Problems at the following link:
Click here for Population Growth Example Maths Questions
World Population Growth
Although Total Population is dramatically increasing, the actual Percentage Rate of world’s population growth is slowing down.
Throughout the 1960s, the world’s population was growing at a rate of about 2% per year.
By 1990, that rate was down to 1.5%, and by the year 2015, it’s estimated that it will drop down to 1%.
Currently each second, 5 people are born, and 2 die, which means that each second of the day we get an extra three people on the planet.
There is an excellent real time Population meter which ticks over continuously, at the following link:
http://www.worldometers.info/world-population/
Family planning initiatives, an ageing population, and the effects of epidemic diseases such as AIDS, are some of the factors behind this rate decrease.
Even at these very low rates of population growth, the population increase numbers are still staggering.
By 2015, despite a low expected 1% growth rate, experts estimated there would be 7 billion people on the planet.
We actually reached 7 billion people four years earlier than this in 2011.
By 2050, there may be as many as 10 billion people living on Planet Earth.
Can the planet support this population and when will we reach the limit of our resources?
Space on the planet is not the problem, as 7 billion people standing shoulder to shoulder would only occupy the area of the city of Los Angeles.
There are already 23 megacities around the world where people live very large numbers.
The main problem is not space, but an inbalance in food and fuel, with 5% of the earth’s population consuming 23% of the world’s energy.
Or in other words 1/20th of the world’s people using up 1/4 of the energy.
13% of the world does not have clean drinking water, and 40% do not have adequate sanitation and sewerage treatment.
The following video is about the science of overpopulation: how it has occurred and what it means to our future.
The following video shows that Population Growth is not the key part of our problems.
War and Poverty are far bigger problems, than our overpopulation.
The Future of Our Planet
The following one hour video documentary shows that extreme poverty has decreased, especially in Asia.
In fact, for the first time on our history, poverty could be totally eliminated.
Although it is one hour long, it is definitely worth watching.
Related Items
Why Exponents and Indices are Important
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Each day Passy’s World provides hundreds of people with mathematics lessons free of charge.
Help us to maintain this free service and keep it growing.
Donate any amount from $2 upwards through PayPal by clicking the PayPal image below. Thank you!
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Enjoy,
Passy
Image Copyright 2013 by Passy’s World of Mathematics
Welcome to Ocean Mathematics!
Passy’s World is pleased to present a series of lessons all about the Mathematics related to the following aspects of the World’s Oceans:
Mathematics of Ocean Waves
Mathematics of Surfing
Surfboard Design and Geometry
Power Generation from Waves
Mathematics of Tsunamis
Sharks Mathematics
Mathematics of Ships at Sea
These lessons have been assembled as part of a Lecture Presentation at the Mathematics Association of Victoria 2013 Annual Conference in Melbourne Australia.
Links to each of these lessons are given below, as well as a link to the PowerPoint Presentations which were used during the lecture.
Mathematics of Ocean Waves and Surfing
Copyright Image Purchased by Passy’s World from Dreamstime.com
In this lesson we look at some of the Mathematics associated with Surfing.
We cover the Mathematics of Ocean Waves, as well as the mathematics and physics associated with catching and riding a surfable wave. People do not need to think about mathematics when actually surfing; however there are certain mathematical realities which are foolish to ignore!
Click the following Link to access this lesson:
Mathematics of Ocean Waves and Surfing
Surfboard Design and Geometry
Image Source: http://wikimedia.org
This lesson covers Modern Surfboard Design, and its associated mathematical geometry.
First we look at the main types of Surfboards, and then we examine the specific geometries of the component parts. We also look at computerised shaping and manufacture of surfboards, involving 3D Printing technology and Parametric Geometrical Equations.
Click the following Link to access this lesson:
Power Generation from Waves
Image Source: http://2.bp.blogspot.com
In this lesson we look at Wave Energy from the Oceans, and some of the Mathematics and Geometry associated with this “Blue Energy”.
Click the following Link to access this lesson:
Mathematics of Tsunamis
Image Source: http://millicentandcarlafran.files.wordpress.com
This lesson examines the Mathematics associated with the incredible destructive power of Tsunami Waves. It’s all about Equations and Algebra Substitution!
Click the following Link to access this lesson:
http://passyworldofmathematics.com/tsunami-mathematics/
Sharks Mathematics
Image Copyright 2012 by Passy’s World of Mathematics
This lesson covers some mathematics associated with Sharks, such as Levy Walks, Biting Force, and Ratio and Proportion.
Click the following Link to access this lesson:
http://passyworldofmathematics.com/shark-mathematics/
Mathematics of Ships at Sea
Image Source: http://i.imgur.com
This lesson contains some Mathematics of Ships including Archimedes Principle, Stabiliser Wings, the Plimsoll Line, the Bulbous Bow, and a number of other aspects relating to ship design and mathematics.
Click the following Link to access this lesson:
MAV Conference PowerPoints
To download the PowerPoint Presentations from the MAV Conference, click the links below and save the files to your computer.
Mathematics of Ocean Waves and Surfing
Click the link below to download this 12MB PPT File:
http://passyworldofmathematics.com/MAVconfPPTs/IntroWavesSurfPPTv4.pptx
Surfboard Design and Geometry
Click the link below to download this 11MB PPT File:
http://passyworldofmathematics.com/MAVconfPPTs/SurfGeomDesignPPTv3.pptx
Power Generation from Waves
Click the link below to download this 3MB PPT File:
http://passyworldofmathematics.com/MAVconfPPTs/WavePowerPPTv2.pptx
Mathematics of Tsunamis
Click the link below to download this 2MB PPT File:
http://passyworldofmathematics.com/MAVconfPPTs/TsunamiMathPPTv1.pptx
Sharks Mathematics
Click the link below to download this 2MB PPT File:
http://passyworldofmathematics.com/MAVconfPPTs/SharksMathPPTv1.pptx
Mathematics of Ships at Sea
Click the link below to download this 4MB PPT File:
http://passyworldofmathematics.com/MAVconfPPTs/ShipsMathPPTv1.pptx
AMC Teacher Resources
The Australian Maritime College has an excellent set of free resources for Mathematics Teachers.
Among these are some very good mathematical exercises related to Ocean Mathematics.
The AMC is planning to add more online interactive lessons, but as of late 2013 the current the interactive lessons are as follows:
A Study of Similar Vessels (an application of Curve Fitting) – 5mins
Wind Farm Feasibility Study (an application of Probability) – 12mins
Vessel Speeds in Waves (an application of Differentiation) – 5mins
Ship Hydrostatics (an application of Integration) – 12mins
Wave Refraction (an application of Trigonometry) – 12mins
Ocean Waves (an application of Superposition) – 5mins
Scaling Laws (an application of Algebra) – 8mins
Go to the following link to find their Resources page:
http://www.amc.edu.au/why-study-maths
To get the workbooks and other resources, it is necessary to register your school and complete an order form online.
These resources are free to Australian schools, and if you would like to see what some of the workbook exercises look like, then check out the following link:
Click here for AMC Online Maths Workbook PDF
The AMC can be contacted about this program at the following email address: whystudymaths@amc.edu.au
Surf Aid Mathematics Resources
Image Screen Captured from Surf Aid Website
Surf Aid is a not for profit organisation sponsored by Billabong with a kean interest in the preservation of Mentawai in Indonesia.
The Mentawai Islands are a chain of about seventy islands and islets off the western coast of Sumatra in Indonesia.
It is totally free to join the “Surfaid Schools Program”, and all that is required is submission of your email address.
The Surfaid website can be found at this link:
http://www.surfaidinternational.org/schoolsprogram
Downloadable Free Units which are in ZIP and PDF format which could be of use to Mathematics Teachers are the following:
AUS Maths 12-14
Mathematics – Connections through Surfing
Technology
Destination Mentawai Islands
The Economics Of Aid
Crossing The Divide – Primary
These are all free PDF and ZIP file downloads
A Typical example might be the following questions about traveling to surfing locations on various Islands:
“If the motor launch can manage an average speed of 15knots, calculate the journey time between each
location. 1knot is 1 nautical mile per hour. You can assume there is sufficient sailing staff to keep moving 24h
per day. Round up your answers to the nearest day and complete worksheet 5.3.
If you move 10º around the equator how many nautical miles have you travelled?
What angle around the equator (change in longitude) corresponds to 1 nautical mile?”
Another activity we looked at was all about “Planning an Overseas Surfing Trip”.
Eg. Costs, Savings Plan, Items needed and their Cost, and so on.
It was very surfing orientated and might alienate non-surfers and female students, but could easily be adapted to be a group of friends planning a Bali Holiday, and undertaking various tours and activites whilst in Bali.
We will be going through all of the Surfaid Materials, and seeing what could be incorporated into some middle school mathematics, even though the school Passy works at has a tiny minority of students who have ever been Surfing.
We suggest you could easily do the same for your classes.
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Feel free to link to any of our Lessons, share them on social networking sites, or use them on Learning Management Systems in Schools.
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These include items of mathematical interest, funny math pictures and cartoons, as well as occasional glimpses into the personal life of “Passy”.
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While you are there, LIKE the page so you can receive our FB updates to your Facebook News Feed.
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Each day Passy’s World provides hundreds of people with mathematics lessons free of charge.
Help us to maintain this free service and keep it growing.
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Enjoy,
Passy
