(Others include “pi” π = 3.141592653…, which is the circumference of any circle divided by its diameter; and “phi” φ = 1.6180339887…, which is the so-called “beauty ratio“). Both of these numbers are irrational (that is, their decimals go on forever and never repeat).

e is also an irrational number and it has value:

e = 2.718281828459…

The number e was “discovered” by several mathematicians (Oughtred, Huygens, Jacob Bernoulli, Mercator and Leibniz) but they didn’t quite know they had stumbled on it and didn’t know its significance.

There are some curious properties of e, one of which is that it’s the limiting value as n → ∞ of (1 + ¹/_n)ⁿ.

It can also be found by adding the infinite sum:

e = 1 + ¹/_1! + ¹/_2! + ¹/_3! + …

So what is e good for?

It is used extensively in logarithms (which was the only way to do difficult calculations for hundreds of years before calculators came along), exponential growth (of populations, money or drug concentrations over time) and complex numbers (which were used to design the computer or mobile device you are reading this on).

So happy “e” day (February 7th, or 2/7).

[For more information on e, see the MacTutor history.]

Related posts:

1. My infinity’s bigger than yours
2. Dinosaur Mathematics…
3. Nursing Entrance Test – for mathematicians or nurses?

In this Newsletter

1. Math tip (a) – Graphs using free math software
2. Math tip (b) – Math of drugs and bodies (pharmacokinetics)
3. Latest IntMath Poll – math applications
4. Latest from the Math Blog
5. Final thoughts

1. Math tip (a) – Graphs using free math software

The latest IntMath Poll asked readers how they normally draw math graphs.

The response from 1900 users was interesting:

65% said they use paper; 20% said graphics calculator and 15% said computer software.

There are many free online and downloadable graphics programs out there and it surprises me so few people use them to draw their math graphs.

In this tip, I show how to avoid some of the pitfalls of using software to draw graphs. Go to:

Graphs using free math software

2. Math tip (b) – Math of drugs and bodies (pharmacokinetics)

Several people have written asking me to write an introduction to pharmacokinetics. This is the process where the body absorbs and metabolizes drugs (or food, or any chemical).

The math involves diffrential equations, but I think everyone will find it an interesting read. Go to:

Math of drugs and bodies (pharmacokinetics)

3. Latest IntMath Poll – Math applications

One of the most common questions from math students is, "When are we ever going to use this stuff?"

This month’s IntMath Poll asks readers whether they feel they get a good understanding of how math is applied in the "real world".

Please add your vote – you can do so on any page in:

Interactive Mathematics.

4. Latest from the Math Blog

A) Math and color blindness

What’s the best way to present math so color blind people can read it? Are you color blind? I’d love to hear your reaction to this article.

B) Camera purchase decisions – how math helps
A site selling electronics uses math concepts to help customers decide.

C) Friday math movie – George Dyson at the birth of the computer
The story of one of the most important inventions ever.

D) Math graphs on the Web without images
Here’s one way to plot good looking graphs on the Web – use ASCIIsvg.

5. Final thoughts

a. Practice

Everyone tells us to practice math and you will become an expert. Here’s another take on that advice.

Practice does not make perfect.
Only perfect practice makes perfect. [Vince Lombardi.]

b. Biodiversity

2010 is the International Year of Biodiversity. The rise and decline of populations is a very interesting math topic.

What can you do to study – and help – endangered species in your area?

Until next time, enjoy whatever you learn.

Related posts:

1. IntMath Newsletter – Drawing graphs, fear of math tests
2. Math and color blindness
3. Math of drugs and bodies (pharmacokinetics)

Let’s have a look at some of the available math graphing tools.

Graphics Calculators

Here are the Texas Instruments TI-83 (left) and Hewlett-Packard HP 40gs (right) calculators.

Graphics calculators are handy, but the screen size is small and there is usually no scale on the axes. Cost is often prohibitive, at around US$100.

Online Graphers

Here’s a few places where you can graph your curve – for free!

The first one is on IntMath.com, and uses Scalable Vector Graphics (you need Firefox browser, or a plugin for IE).

• Plot your own SVG Math Graphs

The following are also free offerings. They are either Flash or Java applets.

• Wolfram|Alpha
• Flash Equation Grapher
• GraphSketch.com
• GraphApplet (warning – red text on a black background)
• GCalc

Computer Applications

Here are some of my favorites. In each case you need to download and install the software.

• GeoGebra (Free. I wrote a review of GeoGebra which shows how to get started with it.)
• Graph 4.3 (Free. Small download, from cnet.com)
• Scientific Notebook (US$90, but does much more than a graphics calculator)

Some problems with using graphics software

If you don’t have a good idea of what a function should look like before your use a graphics package, you can have all sorts of problems.

Here’s an example of a function which catches out the unaware.

Let’s graph it on Geogebra.

Hmmm – it appears to be empty. Is there an error? What’s going on?

We zoom out a few times and start to see 2 curves. Once again, is there a mistake? Why 2 curves?

I zoomed out because I had a good idea in my head what the graph should look like. Since I couldn’t see either of the 2 arms of this curve, and because of the 30 in the denominator, I knew I had to zoom out.

If I left my graph at that, I would still not have a good idea of what the function looks like. I have not chosen a view that shows the crucial features of this graph.

The default view in Geogebra was too close to the origin (0, 0) to see any of the curve.

Let’s have another go, this time using Scientific Notebook. This is what I get when I try to graph the curve.

I get a vertical line passing through 2 on the x-axis. But notice the scale on the y-axis. The number "4e+09" actually means 4 × 10⁹ or 4 billion. So Scientific Notebook recognizes that there are very large values of y involved in this function, and has shown us the limits of its internal coding.

This time if we zoom in, we can start to see the graph appearing as before. But I need to know to zoom in, otherwise I would miss it altogether.

What was that vertical line that appeared before? Was it a mistake? Why did it disappear when we zoomed in?

Next, we use my SVG Grapher. It’s similar to Geogebra in that its default view misses the curves, but is also similar to Scientific Notebook in that it shows a vertical line through 2 on the x-axis.

When we zoom out a bit, this is what we get.

We are still not showing all the vital features of the graph, and that vertical line is still there. What’s that about?

Next, I tried the Flash Math Grapher. Once again I needed to do some zooming. The x- and y- scales are strange (multiples of 13.01??, and is that the x-axis or the line y = -2.98?) , but at least I can get a pretty good view.

Next, let’s call in the big guns. Wolfram|Alpha gives us the following 2 graphs when we put our function in their search box (and they give us a lot more information about the function).

This time we get intelligent graphs that have appropriate x- and y- scales and actually show the curves. Note the first one does not have a vertical line through 2, but the second one does. Why?

That vertical line through x = 2

When the software draws the graph, it chooses regularly-spaced x-values and substitutes them into the function, plots the resulting dots and joins them.

In the above examples that have a vertical line, it means the software has chosen a value just slightly less than 2 (which will give a very small, negative value for y) and another one just slightly more than 2 (which gives a very large positive value for y).

Here’s an exaggerated version of what it’s doing (using only a very few data points joined by straight lines).

If the x-value chosen is exactly 2, the software will normally just skip that value (since it will return a "division by 0" error). The (almost) vertical line should not be included in the graph, since we cannot have x = 2 (this function is undefined for x = 2 since the bottom of the fraction would be zero).

This gap in the graph is called a discontinuity. It should be a gap – not joined by a line as above.

Some software handles this situation gracefully (like the Wolfram|Alpha example) while in others, you need to either understand why the vertical line is there, or in some cases, you can elect to include discontinuities or not.

Best answer

Here’s probably the best way to display the graph of this function.

Our graph has 2 asymptotes. When a curve gets closer and closer to a line but does not touch it, that line is called an asymptote

The first asymptote is the x-axis and the other is the vertical line, x = 2, which I drew using a dashed line of a different color (since it is not part of the graph).

I have shown the 2 asymptotes clearly and I have also labeled the x- and y-axes.

Here’s the function again for convenience.

I knew the graph was going to involve asymptotes since the x-variable is in the denominator (and we can’t have 0 in the denominator) and also considering when x gets really big, the value of the function will be really small.

If I didn’t know this (from graphing many of them on paper), I would have made a mess of graphing my function on a computer.

Graphing software and the future

Will graphing software change what we do in classrooms? Should it? Is it really necessary to sketch graphs on paper still?

If you can draw a quick sketch of a function on paper, it certainly helps your understanding for many types of math problems. The conclusion from above is that it is certainly worthwhile to have a good sense of what a graph should look like before graphing it using software, so we can manipulate the settings to show the graph properly.

Related posts:

1. Free math software downloads
2. GeoGebra math software – a review
3. GrafEq math graphing software

Preparing a syringe.

The process of pharmacokinetics has 5 steps:

• Liberation – the drug is released from the formulation.
• Absorption – the drug enters the body.
• Distribution – the drug disperses throughout the body
• Metabolism – the drug is broken down by the body.
• Excretion – the drug is eliminated from the body.

Of course, each drug needs to act on the body in a different way. Some drugs need to be absorbed quickly (like nitroglycerin if we are having a heart attack) and preferably eliminated quickly (otherwise toxins build up in the blood). For other drugs, we want slow absorption so we get maximum benefit and don’t lose a lot of it from excretion.

So when your doctor prescribes (say) "take 2 tablets every meal time", this is based on the desirable levels of drug concentration and known levels of distribution, metabolism and excretion in the body.

What’s the math?

When the nurse first administers the drug, the concentration of the drug in the blood stream is zero. As the drug moves around the body and is metabolized, the concentration of the drug increases.

There comes a point when the concentration no longer increases and begins to decline. This is the period when the drug is fully distributed and metabolism is taking place. As time goes on, the drug concentration gets less and less and falls below a certain effective amount. Time to take some more pills.

We can model such a situation mathematically with a differential equation. It has 2 parts – an absorption part and an elimination part. At first, absorption (increasing drug concentration) takes precedence and over time, elimination (decreasing concentration) is the most important element.

We have the following variables:

D = drug dose given

V = volume distributed in the body

C = concentration of the drug at time t

F = fraction of dose which has been absorbed (also called bioavailability)

A = absorption rate constant

E = elimination rate constant

t = time

Absorption part: This depends on the amount of the drug given, the fraction that has been absorbed and the absorption rate constant. It decreases as time goes on. The expression for absorption is given by:

A × F × D × e^-At

Elimination part: The elimination dynamic is affected by the elimination constant, the volume distributed in the body and the concentration left of the drug. The expression for this part is:

E × V × C

For our model, we need to subtract the elimination part from the absorption part (since the absorption part increases the concentration of drug and the elimination part decreases it). Our differential equation is as follows:

We now substitute some typical values for our variables (without units to keep things simple. Note C is a variable, the one for which we seek an expression in t.)

Solving this differential equation (using a computer algebra system), gives the concentration at time t as:

C(t) = 533.3(e^−0.4t − e^−0.5t)

We can see in the graph the portion where the concentration increases (up to around t = 3) and levels off. The concentration then decreases to almost zero at t = 24.

Pharmacokinetics is yet another interesting “real life” application of math.

[Based on: A First Course in Pharmacokinetics.
Photo credit: Syringe]

Related posts:

1. Intmath Newsletter – Graphs, pharmacokinetics, color blindness
2. The melting Arctic – a disturbing application of math
3. Today is “e” day

Color blindness is not evenly spread across different ethnic groups. Around 8% of Caucasian men, 5% of Asians, and 4% of African males have the most common type of color blindness — the inability to tell the difference between red and green.

As an example of the difficulties faced by those who cannot distinguish red and green, here is how a red apple is perceived by a deuteranope (someone with the most common red-green color blindness) .

Red apple as perceived by a normally sited person (left) and by a color-blind person (right).

Now we see how a green apple is perceived by a deuteranope.

Green apple (left) and as seen by a deuteranope (right).

As you can see, there is very little difference in the perceived color for the 2 apples and therefore it must be quite a challenge to go grocery shopping. We distinguish many vegetables (including the quality of those vegetables) by their color.

Note that color blind people see some color (it’s not blank or invisible), but they don’t distinguish green and red.

Check how color blind people see your images and text

I obtained the apple images on the right above using the excellent facility at VisCheck. You can upload any image and see how it will look to people with different kinds of color blindness.

Colors of the Rainbow

Here are the rainbow colors, with the way a color blind person would see them.

The Ishihara Color Test

Shinobu Ishihara published tests for color blindness in 1917. They consist of circles with numbers embedded using different colors, as follows.

Note: If you have trouble seeing numbers in the above plates, it does not necessarily mean you are color blind. Each computer monitor has different settings and capabilities. Color blindness tests should only be done using paper print-outs.

Color blindness and learning math

Let’s now look at the implications of color blindness for math students.

Here’s a pie chart that could be used as part of a statistics lesson. The colors are unsaturated, making it difficult to read.

And now for the version that a color blind person sees.

What if the teacher asked a question like "What percent is represented by the orange sector?" The brownish colors are almost indistinguishable and this would make interpretation of this chart quite difficult.

Notice that the order of the items in the legend does not follow the order of the sectors of the pie chart, so there is a double whammy for anyone trying to figure it out.

What colors should we use?

According to Okabe and Ito in the very useful article Color Universal Design, the following are good practices to follow when developing materials.

Principles of Color Universal Design

1. Choose color schemes that can be easily identified by people with all types of color vision.
2. Use a combination of different shapes, positions, line types and coloring patterns, to ensure that information is conveyed to all users including those who cannot distinguish differences in color.
3. Clearly state color names where users are expected to use those names in communication.

No matter what the background color of your presentation, Okabe and Ito advise to avoid red, orange, yellow, yellow-green and green, if such colors are important for distinguishing information. That last bit is important – you don’t need to avoid these colors if they enhance the visual appeal for normally sighted people.

A lot of presenters use color for emphasis, but it can be lost on some viewers.

For example, the following could possibly look all black and so the emphasis is lost.

Dark red characters amongst black text

The following at least will be clearer.

Dark blue characters amongst black text

But please don’t use blue too often on Web pages, since this is traditionally the color for links and should be avoided.

Back to the Pie Chart

Let’s go back to our pie chart and try to improve it. Here’s the chart again with better, more saturated colors. I have used magenta (equal amounts of blue and red, or #ff00ff in color hex) for one of the sectors. Note it looks light blue to a color blind person. Also observe how dark red (#c00000) and light green (#00ff00) appear.

Pie chart with saturated colors (left) and how it appears for a color blind person (right).

It’s better, but far from perfect.

The legend (the dots on the right of each image) adds extra cognitive load for everyone. If we label each sector as follows, it is easier to read and understand.

Line Graphs

Here’s a graph from a page I wrote illustrating Vector Addition. (We’re adding 2 vectors, A and B, and the resultant vector is R.)

I chose primary colors (blue and green for A and B), and "red" for the "resultant" (R), which is fairly commonplace. But look how disastrous it is for a color blind person:

In this case, the vectors are clearly labeled, but the colors do not help at all.

I changed the script to replace green with black. I’m still using red (in the original) for the resultant vector, but I think you’ll agree all colors are distinguishable now.

Redundant Coding

Here’s another kind of line graph that’s found in statistics. It shows a projection of the population by age for the USA.

Here’s how it looks to a color blind person.

As we’ve seen in the other examples, it’s better to label the lines in a chart (rather than use a legend), and to use different patterns (square, triangle, cross, etc) on each line. This is called redundant coding, since we are differentiating the information in more ways than one.

Here’s a better version of this graph, with redundant coding.

I didn’t change the colors (since that would have produced limited benefits).

It’s also best to use thick lines when drawing graphs, since thin ones are more difficult for color blind people to see.

Avoid Certain Color Combinations

Even normally sighted people have difficulty reading certain color combinations. Particularly on Powerpoint slides, it is irritating trying to read red or dark blue text on a dark background, something like this:

Bad enough for normally sighted people, but disastrous for color blind people, as follows.

The usual rules apply for Powerpoint slides – use highly contrasting colors – the best is black on white. Here’s the same text in a form which is easier for everyone to read (black on white):

Conclusion

We can’t assume everyone in an educational setting can see things the way we see them (in fact, that’s almost always the case on many levels, not just color perception!).

Proper labeling of graphs, charts, equations and avoiding indistinguishable colors will certainly help improve readability for all learners – not just the color blind ones.

I realize now I need to go back and fix a lot of my own graphics so they are more readable.

Don’t get me wrong – still use lots of color in your presentations, but keep in mind the needs of those 5% or so of readers who are color blind.

Please comment: Are you color blind? I’d love to hear your feedback or corrections on anything I have written here.

Image Credits

Red apple: CC BY 2.0

Green apple: CC BY-SA 2.0

Ishihara Plates

Population Projections

Related posts:

1. Intmath Newsletter – Graphs, pharmacokinetics, color blindness
2. Making math accessible for the blind
3. Friday Math Movie – Mixed Feelings (Vision through the tongue)

You end up on cnet’s digital camera reviews or perhaps dpreview. Both sites give you good information, but after you’ve spent hours trying to compare makes, models and prices, it becomes overwhelming.

Math to the rescue.

At Retrevo, you can see a visual representation (using a familiar 2-coordinate system), comparing price (horizontal axis) and value (vertical axis) for a large range of cameras.

Here’s a screen shot. The dots represent cameras, and the flags are for the best value cameras (currently a Lumix camera is regarded as best value).

So if you want a cheap camera that is great value, choose one in the top left corner. If you’re going for something classy, avoid the duds (expensive but poor value) in the bottom right quadrant.

Over in the printer section, we see a rather mathematical way of representing the developmental position of a product (this is the Epson Artisan 810 printer, which is “buzzing” currently):

Apparently the idea is to buy something on the ascendancy, rather than a “has-been”.

Expect more of this type of analysis on Web sites, particularly high-traffic ones selling electronics products.

A picture is worth a thousand words.

Related posts:

1. Canon woes
2. Review of Top Math Sites
3. Canon i455 Printer

Historian and philosopher of science, George Dyson, talks about early computer concepts (including those by mathematician Leibniz), ENIAC (Electronic Numerical Integrator And Computer) in the 1940s, the use of computers to create atomic bombs, the huge frustrations involved in early computers, the mouse and punch cards.

It’s interesting when he talks about the computer “evolving” into a new kind of “symbio-organism”.

Related posts:

1. Friday Math Movie – Toys that make worlds
2. Friday Math Movie – George Bush fuzzy math
3. Friday Math Movie – the Web’s secret stories

In this Newsletter

1. What’s new on IntMath; and plans for 2010
2. Math tip – Length of an arc
3. Subscribe, connect & communicate with IntMath
4. Latest from the Math Blog
5. Final thought – Goals

1. What’s new on IntMath

(a) Math graphs without images

Need to draw the graph of a function?

I’ve added some new pages where you can see math graphs using scalable vector graphics rather than images (like .gif, .jpg or .png). You’ll see a different graph each time you load the page (not like images, where it’s the same each time you see it).

• Adding vectors: This page in the Vectors chapter produces a different example each time you re-load it: Adding Vectors.
• User-generated graphs: You can plot your own math graphs on this page: Plot your own SVG Graphs

[Please use Firefox, Chrome, Safari or Opera browsers. Internet Explorer requires Adobe's SVG Viewer plugin for this to work.]

(b) Faster IntMath

Slow-loading Web pages irritate everyone. I did some tweaks so that all pages on IntMath load faster. I hope it improves your enjoyment of the site!

(c) New in the Newsletter

When you subscribed to the Intmath Newsletter, I gave you an opportunity to indicate the topics you’d like me to cover.

Well, the list of requested topics is now very long (several hundred topics!). I can’t cover them all, but I’ll do my best.

Also, there is quite a broad range of people reading this Newsletter. Most are students, some of whom are beginners studying algebra, while others are doing more advanced topics, like trigonometry, logarithms and graphs, while others are studying calculus.

My attempt in each Newsletter is to cover a broad range of topics and hopefully you find something interesting in each one.

For future Newsletters, I’ll still aim to have 2 Math Tips, with the first one being more “general” in nature and applicable to just about anyone at any level of mathematics; while the second one will be aimed at those of you who have been studying math for a longer time.

(d) Plans for 2010

I have lots of interesting new things in store involving IntMath during 2010. Watch the Newsletters for announcements!

2. Math tip – Length of an arc

A reader who works for a glass company (true story) wrote to me recently and asked how to solve the following.

I’ve got to make a window with a curved top. The width of the frame is the same all round, including the part around the curved portion.

What’s a formula for the length of the inner arc of the curved portion?

The required length is labeled HI in the diagram.

(This is not quite a “Norman window”, would have a complete semi-circle on top.)

This is a typical “real life” question, in that we don’t have a lot of information to go on, so we’ll need to make some assumptions.

Try it yourself first. The solution involves trigonometry and the use of radians (a better alternative to degrees).

Then check out the full solution.

3. Subscribe, connect & communicate with IntMath

Here are a few ways to keep up with the latest on IntMath:

• Subscribe to the blog so you are alerted every time there is a new article. (you can use the blog feed in your iGoogle or MyYahoo page).
• Get the “daily Math Tweet” on Twitter: IntMath on Twitter
• If you want to connect with me on Facebook, please say you’re a subscriber to the IntMath Newsletter. Otherwise, I won’t know who you are.
• I love to get your feedback. You can comment on any article in the blog or ask your questions in the Comments section in IntMath..

4. Latest from the Math Blog

1) New names for the days of the week

Perhaps it’s time to move away from the ancient names used for the days of the week?

2) Classic math mistakes

Displaying posters of math mistakes can actually help students to learn.

3) Friday math movie – Pump up your brain

Looking for a sure-fire way to boost your math scores? Check out this video that helps you to pump up your brain.

4) Fair slicing of the pizza – using math

Not all pizza gets sliced neatly through the middle. How can we ensure a fair distribution of slice area?

5) Friday Math Movie – the Web’s secret stories This is one artist’s visualization of the emotions of the Web.

5. Final thought – Goals

The beginning of a new year is a good time to reflect on what we achieved last year and what is in store for 2010.

There are 3 types of people. Are you one who:

• Makes things happen?
• Watches things happen?
• Wonders what happened?

In every math class, there are those who ensure their learning is efficient and enjoyable. They make it happen by being organized, reading ahead, listening in class and practicing what they learned before it fades away.

Then there are those who regard math as a “spectator sport” and let the teacher do all the work — then wonder why they aren’t doing very well.

Finally, there are those who have no idea what is going on. They are easily distracted and have no idea how to answer if they are asked a question.

So what are your math goals for 2010? Will you be someone who makes things happen and therefore improve your chances for success?

Until next time.

Related posts:

1. IntMath Newsletter – Radicals, Integrator and Goals
2. Arc length for the inner curve of a window
3. Intmath Newsletter – Graphs, pharmacokinetics, color blindness

I’ve got to make a window with a curved top. The width of the frame is the same all round, including the part around the curved portion.

What’s a formula for the length of the inner arc of the curved portion?

The required length is labeled HI in the diagram.

(This is not quite a “Norman window”, since it’s not a semi-circle on top.)

This is a typical “real life” question, in that we don’t have a lot of information to go on, so we’ll need to make some assumptions.

Solution – Example

Let’s consider a plausible example first. We assume the curves are arcs of circles. (They looked circular in the question. If they are not circles we could adjust later, but for the sake of the example, we’ll stick to this reasonable assumption.)

Let the total frame (length CD) be say 4 units wide and the edges of the frame be 0.3 units wide. I pick a point P in the center (horizontally) of the frame (I choose point P (2,1)), and draw 2 concentric circles, 0.3 units apart. The outer one is 5 units, the inner one 4.7 units. I could have chosen any radii for my concentric circles, of course, as long as the inner and outer radii differ by 0.3 units.

If I can find angle θ, it will be straightforward to find the arc length HI.

First, we find angle α by using the right triangle PMI.

cos α = 1.7 / 4.7 = 0.3617

Using the inverse ratio, we get that

α = arccos (0.3617) = 1.2007 (radians, of course. If we need degrees, it equals 68.79°)

[Why radians? They are more commonly used in science and engineering than degrees, and are best for this problem. For more, see Radians.]

Now we observe that 2α + θ = π (180°), since they lie on a straight line.

So angle θ = π – 2 × 1.2007 = 0.7402

To find the arc length HI, we just apply the arc length formula

s = r θ

(See arc length formula)

s = 4.7 × 0.7402

= 3.4789

So the required arc length is 3.48 units (correct to 2 decimal places).

General Solution

Let the total width (window and frame) = 2x, giving x for half the width.

Let the edges of the frame have width w.

The circles have radius R (outer) and r = R – w (inner).

Angle α = arccos ((x – w) / r)

Angle θ = π – 2 α (in radians)

Arclength HI is given by

s = r θ

= r × (π – 2 (arccos ((x – w) /r)))

This is the formula required by the glass manufacturer.

Related posts:

1. Intmath Newsletter – Length of an arc, what’s new, goals
2. Hubble math
3. Curved spacetime simulation

I’m always on the lookout for alternatives to images for Interactive Mathematics.

Scalable vector graphics offer many advantages over images. A scalable vector graphic is a small text file that your browser (if you are using a good browser) renders as an image. It solves each of the issues raised above. Readers can easily change the graph to whatever math function they want to see, and such graphics involve very small file sizes.

One easy way to implement of scalable vector graphics on a Web page is to use ASCIIsvg. Below is an example, showing the graph of y = sin e^x.

Note 1: You need to use Firefox browser (Chrome and Opera also work) to see the graph below. It will only work in Internet Explorer if you have Adobe’s SVG Viewer plugin.

Note 2: You need to view this page in the original blog post. It won’t work if you are viewing via an RSS feed in your aggregator.

You can mouse over the graph and it shows the coordinates of the cursor position. You can use this to determine particular points on your graph (for example, where the curve cuts the x-axis).

You can double click on the graph to see the code that was used to create it. You can also change parameters in this code then re-draw the graph. (Click the “Update” button after you’ve made your changes.)

If you can’t see a graph above and don’t know what I am talking about, this is a screen shot. If you view this article using Firefox (or Chrome or Opera) browser, you’ll see something like this:

Screen shot of SVG plot

How does it Work?

The graph uses the native SVG capability of (good) browsers (like Firefox, Chrome and Opera). SVG stands for scalable vector graphics, which as mentioned above, produces high quality images usually with very small file sizes. (You can learn more about vector-based art here: Vector Art).

For the graph above, the only code used on the Web page is this:

\begin {graph} width=400; height=300; xmin=-1.5; xmax=3.5;
ymin=-1.5; ymax=1.5; xscl=1; axes(); plot(sin(e^x));  \end{graph}

SVG is very powerful, but can be difficult to obtain the graphs you want. So Stephen Jipsen of Chapman University developed ASCIISVGsvg which is a javascript interface for SVG. That is, it makes our lives easier.

Earlier I wrote about the related solution ASCIIMathML (see Enter math in emails, forums and Web pages using ASCIIMathML). ASCIISVGsvg is included in some versions of ASCIIMathML.

How do I set up ASCIIsvg on my own Web page or blog?

You just need to download 2 files from ASCIIsvg main page:

1. d.svg (this is a very small file that the “embed” tag uses)
2. ASCIIsvg.js (this is the javascript that talks to the SVG rendering capability of your browser)

Then, add the code to embed an SVG graph, which will be something like this:

<embed width="117" height="117" src="d.svg"
script='initPicture(-2,2)
axes()
stroke = "blue"
plot(2x^2 - 3x)
'>

Finally, add a call to the javascript file (best at the bottom of your HTML page) something like this:

<script type="text/javascript"
src="http://mysite.com/ASCIIsvg_min.js"></script>

Advantages of Drawing Graphs using SVG

1. The old way: I drew nearly all of the graphs on Interactive Mathematics using mathematics software (either LiveMath, Scientific Notebook or similar) and then I took a screen shot using an image editing software. I then saved the image and uploaded it to the server. This is a many-step process that is fiddly and inconvenient, especially if you want to change anything on the graph.

   But SVG-based graphs are much easier to work with. You just change a bit of simple code and you get your new graph right there on the page.

2. Another key advantage of SVG graphs is the small file size, since it just uses a small bit of simple code. This is good for you (it downloads quickly) and me (it reduces bandwidth costs).
3. Readers can plot their own graphs on the Web without having to create images. (See how below.)
4. You can print high quality graphs from your browser

Disadvantages of Graphs using SVG

1. It is not (yet) cross-browser, since Internet Explorer does not render SVG. This is a big limitation, but since IE is now less than half of the browser market, maybe it won’t be a problem for much longer. (Well, we live in hope. )
2. You need the javascript to run on the Web page. So as I said above, you can’t see the graph if you are reading this via a feed.
3. Despite what I said above about file sizes, your browser will download the ASCIIMathML javascript, and this is not so small at 150kB. But if you use ASCIIsvg only and use file compression on your server, it comes down to 7 kB only, which is less than many images on the Web. Of course, your browser will cache it so it is a one-time download.

My Examples of Graphs using SVG

There are a few versions of ASCIIsvg floating around. I first started with the ASCIIsvg which is inbuilt into ASCIIMathML, but I found it was quite limited. (I spent too long trying to figure out how to change parameters and gave up.)

I settled on (pure) ASCIIsvg since it was much more transparent. A big downside is it uses the “<embed>” tag, which is not HTML-compliant.

Adding vectors: I produced a page in the Vectors chapter: Adding Vectors which produces a different example each time you re-load the page.

User-generated graphs: You can plot your own graphs on this page: Plot your own SVG Math Graphs

Resources for ASCIIsvg

You can obtain ASCIIMathML from here: ASCIIMathML.

If you only want graphs and not the whole math rendering engine, you can get ASCIIsvg here: ASCIIsvg main page

Get your questions answered here: ASCIIMath Google Group

More examples of graphs using SVG: SVG Gallery

Related posts:

1. Graphs using free math software
2. Enter math in emails, forums and Web pages using ASCIIMathML
3. Friday math movie – Trigonometric Graphs