IntMath will be hosting the next Carnival of Mathematics, a collection of recent blog posts about math.

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]]>To submit an article, go to:

As it says on the submission form:

The Carnival of Mathematics accepts any mathematics-related blog posts: explanations of serious mathematics, puzzles, writing about mathematics education, mathematical anecdotes, refutations of bad mathematics, applications, reviews, etc. Sufficiently mathematized portions of other disciplines are also acceptable. We also accept YouTube videos, and non-blog based content, as long as it’s new and has been recently posted online.

The most recent carnival was hosted by Mike at Walking Randomly. It covered the full gamut from limits to Ninjas and 3D printed geometry to mathematically-related underwear. See

All the previous carnivals can be found on Aperiodical. See:

See you back here in mid-September for edition #114!

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]]>The post Context Free math-based art appeared first on squareCircleZ.

]]>Context Free is software that you can use to generate fascinating art work. You give instructions using code called a "grammar". It’s quite a neat way to bring together math, art and programming, which are key STEM skills.

The Context Free Art site provides downloads of the (free) software (Mac, Windows and Linux), and also a gallery of around 1000 works of art generated using the program.

Everything is released using a Creative Commons non-commercial license, so you can use the images in your own work. You can modify other people’s code to create your own materpieces (but give credit to the original, of course).

Here are some of the output images from the gallery:

Drawing these images involves some interesting math topics, including:

- Trigonometry
- Logarithms
- Matrices
- Vectors
- Geometry
- Number theory
- Complex numbers

This one is called the Lorenz Attractor, by thijs:

To give you an idea what is involved, this is the code thijs wrote to produce the above shape:

startshape Lorenz_Attractor //1600*900 CF::Background = [b -0.9 sat 0.2] CF::Size=[s 60 33.75] CF::ColorDepth = 16 CF::MinimumSize= 0.3 CF::Impure = 1 A=10 B=8/3 C=28 d=0.002 vector3 D(vector3 V) =V[0]+d*( A*(V[1]-V[0]) ) ,V[1]+d*( V[0]*(C-V[2])-V[1] ) ,V[2]+d*( V[0]*V[1]-B*V[2] ) vector3 M(vector3 V1,vector3 V2) =(V1[0]+V2[0])/2 ,(V1[1]+V2[1])/2 ,(V1[2]+V2[2])/2 //------------------------------------ shape Lorenz_Attractor { V0=(-12.5, -17.4, 26) Next(0,V0)[r 100] } shape Next (i,vector3 Vi) { if (i<320000) { Vn=M(Vi,D(D(Vi))) Plot (i,Vi,Vn) [] Next (i+1 ,Vn) [] } } shape Plot(i,vector3 V0,vector3 V1) { P=V0[0,2] Q=V1[0,2] Z=V0[2] H=45+15*sin(i/3) Line(0.05,P,Q) [z Z b 1 h H sat .4] Line(0.15,P,Q) [z Z b -1 a -.950] Line(1.50,P,Q) [z Z b -1 a -.999] } path Line(S,vector2 P,vector2 Q) { MOVETO(P) LINETO(Q) STROKE(S,CF::RoundCap)[]

Here's a useful list of resources by Mikael Hvidtfeldt Christensen: Generative Art Links, containing links to some interesting software for creating several different types of math art, including 3D fractals, and there's also some interesting blogs.

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]]>The updated applets on the Graphs of tan, cot, sec and csc page now run on mobile devices, albeit rather slowly. One of the most popular pages on IntMath.

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]]>To make the page more mobile-device friendly, I updated it so now the interactive graphs use JSXGraph, rather than Flash.

The page explains how to sketch these trigonometric curves, mostly by considering the denominator (the bottom) of the function expression.

Each of the curves tan *θ*, cot *θ*, sec *θ* and csc *θ* has a discontinuity, or "gap" in the curve. For example, tan *θ* is defined by:

Since cos *θ* has value 0 at *θ* = … −3π/2 −π/2, π/2, 3π/2, 5π/2,…, it means tan *θ* will not exist for those values of *θ*. The result is there will be "gaps" in the curve at those points.

Either side of those gaps, the curve heads off to plus or minus infinity. Here is the graph of *y = *tan *θ*:

The page has animations that demonstrate where the values of the curves come from.

Here’s a screen shot of one of the animations:

On a tablet, the animations may run a little slow. After all, we’re asking the processor to perform a lot of calculations! Alo, the screen gets re-drawn multiple times during the animation, which also slows things down.

As a result, you can only animate one of the graphs at a time.

The link again:

Graphs of tan, cot, sec and csc

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]]>Here's some background on the new Riemann Sums applet I added to IntMath. You can see how we find areas under curves using rectangles.

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]]>I recently added a new Riemann Sums Applet to IntMath which demonstrates some of the ways it can be done, without calculus.

Here is some background to the concept.

For centuries, mathematicians tried to find efficient ways of calculating rates of change. They needed to solve a wide range of practical problems, including building faster ships, developing more accurate weapons and predicting the positions of the stars and planets for better navigation.

They were also interested in the reverse problem. That is, they may have known the rate of change, but they wanted to know the fundamental relationship between two changing quantities. For example, they may have known the velocity of an object at different times, and they wanted to know the position of the object in space during those times.

Since the age of the ancient Greeks, there were good approximation methods for such problems.

A Riemann Sum is an approximation of the area under a curve (that is, between a curve and an axis). It’s named after Bernhard Riemann, the German mathematician who worked on this problem in the mid-19th century.

The principle behind Riemann Sums is simple, and was known since the time of Archimedes. (See Archimedes and the area of a parabolic segment.)

We just divide the area into smaller pieces using simple geometric shapes for which we can easily find the area, like rectangles.

If we find the area of each rectangle, and add those areas, we’ll get a good approximation for the area under the curve. If we take more rectangles, the error will reduce, and we’ll get a better approximation.

We can place the rectangles in different ways (just above the curve, just below, or mid-way like in the above image).

I recently added a new applet that allows you to explore this concept. You can change the placement of the rectangles, and you can even see how trapezoids and parabolas are used as very good approximations.

You’ll find the new applet here: Riemann Sums Applet

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]]>What is the length of a spiral around a paraboloid? This is an interesting application of integral calculus, involving an environmentally friendly cooker design.

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]]>A reader from Africa, Magnar, recently asked about running a string in a spiral around a parabolic dish, as illustrated in the following image:

Magnar wanted to know how to calculate the length of such a spiral.

There’s an interesting story behind Magnar’s question. He builds parabolic dish cookers in Africa. In this photo, he’s helping to install one of his dishes in a village. The dish concentrates the heat of the sun and the cook pot is placed at the focus of the parabola.

Image source (used with permission)

This is a great use of sustainable energy, it keeps the air cleaner (by avoiding open fires inside huts), and saves women from the backbreaking and time-consuming task of gathering firewood every day.

Let’s look at some of the math behind this problem.

A paraboloid is the 3D surface resulting from the rotation of a parabola around an axis. The equation of a simple paraboloid is given by the folmula:

*z* = *x*^{2} + *y*^{2}

The surface generated by that equation looks like this, if we take values of both *x * and *y* from −5 to 5:

Some typical points on this curve are (0,0,0), (1,1,2), (-2,3,13) and (3,4,25). That last point is on the top lip of the surface in the 3-D graph given above, at the front, facing us. (In this graph, the *x*- and *y*-axes axes are equally scaled, but the *z*-axis is not.)

We can also represent this paraboloid using parametric equations with variables *t* and *r*, as follows:

*x = r *cos(*t*)

*y = r *sin(*t*)

*z* = *r*^{2}

We could write this as an ordered triple, like this:

(*r *cos(*t*), *r *sin(*t*), *r*^{2})

We only need to let *t* take values from 0 to 2π (otherwise we just "paint" the surface more than necessary) and *r* takes values from 0 to 5 if our aim is to create the same curved surface as shown in the above graph.

Let’s get back to the original problem. We want to create a spiral around the surface of the paraboloid. We modify the earlier parametric equations to get a curve rather than a surface, like this.

*x = t *cos(*t*)

*y = t *sin(*t*)

*z* = *t*^{2}

Once again, we could write this as an ordered triple, as follows:

(*t *cos(*t*), *t *sin(*t*), *t*^{2})

We no longer create a surface with this expression. Instead, it will be a curve.

As the variable *t* takes various values starting from *t* = 0, it generates a spiral around the 3-D surface.

Let’s see how, starting with a simple case.

Let’s consider the case where we fix the curve so it is in the *x-z* plane only (it’s not actually spiralling in this case, to make things simple.) We write this as:

(*t*, 0, *t*^{2})

The graph of this is part of a parabola, starting at (0,0,0) and extending to (20,0,400), as shown.

What is the length of this simple curve?

We use a result from calculus, which is based on the Pythagorean Theorem.

The arc (or curved) length is given by:

(See Arc Length of a Curve and Arc Length of a Curve in Parametric or Polar Coordinates for background on these formulas.)

In this example where *x* = *t* and *z* = *t*^{2},

and

Also, in this example we have the end point *t*_{1} = 20.

Substituting gives:

(I used a computer algebra system to find this integral.)

This seems quite a low firgure for the length, since the *z*-value at the end of the curve is 400, and going around the curve should add more than just over 1 unit to the length.

However, the above graph does not have equally-scaled axes, whereas the following one does.

We see that a parabola is "almost" straight, and a curve length of just over 400 for this curve is reasonable.

Let’s now see the effect of those trigonometric terms we saw before.

We now plot (*t *cos(*t*), *t *sin(*t*),* t*^{2}) for *t* = 0 to *t* = 20. (Of course, *t* is in radians.)

The start of the curve, when *t* = 0, is at the point (0,0,0). The point where the curve finishes, when *t* = 20, is:

(20 cos(20), 20 sin(20), 20^{2}) = (8.162, 18.259, 400).

For interest, looking from above (down along the *z*-axis), the above spiral looks like this:

The arms cut the *x*-axis evenly at 2π = 6.28, 4π = 12.57, and 6π = 18.85.

You can see the curve ends near (8, 18) as we claimed before.

We extend the 2-dimensional case above now that we are working in 3 dimensions. In general, the length of an arc when using the parametric terms ((*x*(*t*), *y*(*t*),* z*(*t*)) is given by:

Substituting in our expressions for * x*, *y* and *z*, we have:

Now to examine the solar cooker problem.

Considering one of Magnar’s cooker designs, the parabola has a cross-section which is quite flat compared to the paraboloid we drew above. (Like most parabolic dishes, they are using flat panels to approximate the curved surface.)

In fact, the top lip passes through the point (25,45) in that design. The general form of a parabola can be written:

*z = cx*^{2}

Substituting (25,45) gives us .

We now plot the curve

from *t* = 0 to *t* = 45.

The curve starts at (0,0,0) and the top point is (23.6, 38.3, 45). We see there are about 7 spirals.

Here it is from the side:

And next the view from the top shows we finish near *x* = 24 and *y* = 38:

We have around 7 turns in the above spiral, but we may need more or less. If we want to achieve more spirals within the same *x-, y-, *and *z-*constraints, we need to multiply the variable within the trigonometric terms by a number.

Here is the case when we multiply the variables in the trig terms by 5. That is:

We get around 35 spirals this time:

Here it is from the side.

The length of this spiral is:

Similarly, if we multiply those variables by a number less than 1, we get less spirals. Here is the case when we multiply by 0.3:

This time we have just over 2 spirals (which is similar to the situation we had in Magnar’s first diagram at the top) and our length is:

This has been an interesting application of math where 3-D geometry and integral calculus are being used to examine a good use of solar power. We need to make more use of the huge amount of energy available from the sun, and solar cook pots are a great way to do so.

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]]>The post From brain injury to math genius: Jason Padgett appeared first on squareCircleZ.

]]>Jason Padgett was attacked by 2 men one night and savagely beaten. He ended up with severe concussion and an amazing new ability to "see the world through the lens of geometry".

He began to create intricate geometric art, like this image:

See the full story at Discovery.com: Brain Injury Turns Man Into Math Genius. Don’t miss the gallery of his art while you’re there.

Padgett’s story reminded me of the work of Alan Snyder, the Sydney-based researcher who is able to temporarily "turn on" savant-like mathematical abilities in test subjects. (See Switching on your inner Rainman: Enhancing Creativity).

Snyder uses magnetic pulses through the brain to release what he believes are math and creative abilities inherent in all of us.

Padgett wouldn’t change his new abilities if he could.

"It’s so good, I can’t even describe it," he said.

It’s fascinating that Padgett has discovered the joys of math patterns, but it’s very sad it took a violent event to bring it about.

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]]>Google tries hard to provide super-fast Web pages, and uses calculus in the process.

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]]>Google’s success as a company was built on **optimization**. They’ve put a lot of effort into making everything they do super fast.

Google wants all pages to be fast, so they provide a free Web page speed test facility, WebPageTest.org. You can test any page there and see why it is so slow (or fast). It’s great for those of us who are keen on speeding up our sites.

Of course, there are many things that will slow down a page, including slow Net connections, a slow server, size of the page (in kb), slow download speeds and slow browser processing speeds.

WebPageTest provides various test result measures, including connection time, download times and a metric called **Speed Index**.

Speed Index is an indication of how quickly the page elements appear on a page. In some cases, we see just a small part of the page and then see the browser "spinner" indicating more is to come (we are likely to leave in such cases). On other pages, we may see most of the page quickly, but the spinner indicates some more is coming.

Obviously the second cane is a better user experience, so it should get a better Speed Index score.

The Speed Index documentation explains how they calculate the score. It involves finding the **area under a curve**, which is a concept we learn when first meeting integral calculus.

They give a formula:

where

*T* is the "visually complete" time (total time to load and process the page); and

*VC*/100 is the "percent visually complete".

The areas are calculated at 0.1 s intervals (or similar) and added to produce the final index.

The area represented by the integral is the darker blue area in this chart, which represents a fast Web page (most of the content is visible quickly and the last 10% or so appears later).

[Image credit: WebPageTest]

So in fact, we are finding the **area between 2 curves,** the top one is the constant value 1 (or 100%), and the other is the VC curve.

Some closing ironies:

- The documentation page referred to above took around 27 seconds to "visually complete" status when I opened it today. They haven’t even optimized images on that page to reduce the download time.
- Google is not always quick. Gmail has many elements and takes a long time to load when first opened.

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]]>The post Friday math movie: 5 Quick Math Tricks for Filmmakers appeared first on squareCircleZ.

]]>Here are some math-related tips for better movies.

He discusses the f-stop scale, what an appropriate shutter angle is, the 180-degree Rule and Rule of Thirds, and how to read a histogram.

Actually, there are plenty of potential interesting math lessons arising from this. How about get students to make relevant videos about the math they are using, while making videos!

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]]>The post Friday math movie: Falling in love with primes appeared first on squareCircleZ.

]]>He mentions the largest current prime, which is around 12.5 million digits long.

Monster, indeed.

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]]>Here's some math background behind the recently updated pulley applet.

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]]>As I developed it, I used quite a bit of interesting math.

I get quite a few mails from people asking how I develop these applets, so here is some mathematical background.

The 2 circles are built using the basic formula for a circle:

The left-hand circle has center (0,0), (I could have placed it anywhere on the *x-y* plane, but (0,0) seemed simplest), and the radius it has at the start is 5. So its equation is:

The right hand circle has center (15,0) and radius 2 at the start.

If the circle has center other than the origin, say (*h,k*), we use this formula:

So the equation of the right hand circle is:

[JSXGraph allows us to easily draw circles with different centers and different radii (without resorting to the formulas), but I still needed to make use of the formulas for later steps.]

I have hidden the axes for the applet to make it cleaner.

The red dot follows the belt around. It travels through 2 circular arcs, and 2 straight line segments.

I needed to find the intersection point of the arcs with the straight line segments. This could be done using either Non-linear systems of equations or via other geometric constructions.

Of course these intersection points change when the radius of each wheel changes.

I assumed no gravity was operating on the belt, and that it is (very) tight.

The red dot follows 2 line segments. I needed to find the equation of those segments. So I needed to use the basic equation for a straight line with slope *m* and *y*-intercept *c*:

To find the actual values of *m* and *c*, I needed to use the following formula, once I found the intersection points and :

The red dot travels around 2 circular arcs, and there are points (the circles) on the pulley wheels that rotate around a center as well.

I needed to use the following parametric equations, where *r* is the distance from the center to the rotating point, to achieve this:

As time goes on, the angle *θ* changes (it’s in radians, of course), and the point sweeps out a circle. The starting point is (*r*, 0) when θ = 0.

Of course, the whole point of the applet is to demonstrate the relationship between linear and angular velocity, one of the applications of radian measure. Our basic equation is:

*v = rω*,

where

*v* is velocity in meters/second (m/s)

*r* is radius in meters (m)

*ω* is angular velocity in radians per second (rad/s)

I find I understand new formulas more easily when I can see the units.

Pulleys are used to make our lives easier (e.g. gears on a bicycle, or for lifting weights) and they work when there is a difference in the radius between the 2 pulley wheels. So we may need to pedal faster on a bike when in a low gear, but the advantage is we don’t have to push so hard.

The applet shows the difference in angular velocity between the larger wheel (lower angular velocity) and the smaller wheel (higher angular velocity).

It turns out the ratio of the angular velocities is the inverse of the ratio of the radii of the 2 wheels.

In programming, we can come across 2 kinds of functions:

- A
**set of commands**that is used over and over (for example there is a function that draws the pulley wheels and the segments, adn this function is called each time you change the radius of a wheel, and the function that moves the red dot and other points around) - The
**mathematical-type function**. In this applet, I make use of sine and cosine functions (as well as inverse sine and cosine) to create the parametric equation points I mentioned earlier.

I also needed to use the concept of shifted trigonometric functions (ones where the curve does not start at 0).

This is actually a relatively simple applet. Programming such applets – and most games – involves a lot of interesting math!

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