Month: March 2017

If you have already created the phyllotaxis data in Fathom, or opened the data created in TinkerPlots in Fathom, you can change the display you get when you drag open a collection box. Under the display tab on the Collection Inspector in Fathom you can set the display properties so that cases no longer show up as uniform gold balls. Setting the display attributes to the values below should give you a growing spiral like the one pictured above.

x = 10x+400
y = 10y+400
image = greyCircleIcon
width = sqrt(r*10)
height = sqrt(r*10)
caption=””

Note that the x and y on the right-hand side of the equations above are the x and y attributes that you defined for the data, and the x and y on the left-hand correspond to the position of the icons. You can experiment with other formulas for width and height—using a slider to provide a variable instead of the number “10” gives more flexibility.

The images below show some of the other spirals you can obtain by varying the angle between the seeds, as mentioned in the previous post.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*

Robert Recorde, author the first English textbook on algebra (published in 1557), chose to give his book the metaphorical title The Whetstone of Witte to encourage people to take up the new and difficult practice of algebra. The metaphor of a whetstone, or blade-sharpener, suggests that algebra is not only useful, but also good mental exercise. In the verse that he included on its title page, he writes,

Its use is great, and more than one. Here if you lift your wits to wet, Much sharpness thereby shall you get. Dull wits hereby do greatly mend, Sharp wits are fined to their full end.

Mathematics, and algebra in particular, according to The Whetstone of Witte is like a knife-sharpener for the brain. Four hundred years later, in his book Mathematician’s Delight (1961), W.W. Sawyer takes up a similar metaphor, suggesting that “Mathematics is like a chest of tools: before studying the tools in detail, a good workman should know the object of each, when it is used, how it is used.” Whether they describe mathematics as a sharpener or other tool, these mechanical metaphors are commonly used to emphasize the practicality and versatility of mathematics, particularly when employed in engineering or science, and suggest that it should be used thoughtfully, and with precision.

An often quoted mechanical metaphor that suggests a more frantic and less precise process of mathematical creation is often attributed to Paul Erdos: “a mathematician is a machine for turning coffee into theorems.”

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*

Building polyhedral models is a nice way to explore a lot of significant mathematics. The above models were made by printing patterns onto card-stock, cutting them out, and gluing them together. For these models, only triangular faces were used. These can give you a wide variety of cumulated (or augmented) polyhedra. The triangular faces are circumscribed to provide tabs that you glue together. You can fold and glue the tabs so that they are inside the models, but it is easier to leave them out, and they look nice this way (I think).
1. Decide on a model that you would like to make, and figure out how many faces you will need.
2. Copy and paste the images below into a document or presentation slide (PowerPoint works well) for printing. Choose the right ones for your model, and fit as many as you can on a single sheet.
3. Print out onto card-stock. Most desk ink-jet printers can take card-stock instead of printer-paper.
4. Cut out the units, fold the tabs, and assemble and glue.
Throughout this process, it helps if you have pictures of the polyhedra that you want to construct. Poly is a nice software package for browsing through families of polyhedra.
I’ve found that it works well to bend the tabs using a ruler, that glue-sticks provide the best gluing, and that it helps to hold the model together with binder-clips while assembling.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*

Phyllotaxis is a term used for the patterns that emerge in the growth of plants. Spiral phyllotaxis is observed in the heads of sunflowers, in pine-cones and pineapples, and in a variety of other plants.

Phyllotaxis is a popular topic in mathematics recreation – it’s interesting in its own right and also related to other perennial favourites, Fibonacci numbers and the golden ratio.

The article Modeling Spiral Growth in a GSP Environment describes how to model phyllotaxis-like patterns in GSP. Although GSP works reasonably well, TinkerPlots or Fathom environments seem to be better suited to capturing this particular model – they make the formulas more explicit and easy to manipulate, and they allow for the data generated to be viewed in a variety of ways. The images above were created by porting this model to TinkerPlots.

As the article suggests, experimenting with the the angle between successive seeds allows you to see different resulting patterns – angles that are multiples of rational numbers create patterns of rays while irrational numbers (actually approximate values) give spirals, or spiral/ray combinations (the rays form as the approximation gets more “rational”). A good choice for approximating actual phyllotaxis patterns is to use tau = (1+sqrt(5))/2 in your angle. Here is a listing for the attributes required to generate the pattern in TP or Fathom. The graph/plot is simply the x attribute on the horizontal and the y attribute on the vertical (in TP these need to be fully separated).

n = caseIndex
base_angle = pi*(1+sqrt(5))
r = sqrt(n)
theta = n*base_angle
x = r*cos(theta)
y = r*sin(theta)

The images shown in this post use a collection of 500 cases or “seeds”. The base angle is 2pi*tau, and the actual angle for a given seed is a multiple of this base angle.

The model is nice looking and easy to build, but it models only the end result of the growth process, not the process itself. It winds new seeds around the outer edge based on a pre-determined angle. A better model would be one that mirrors what is understood to be the underlying phonomena – new seeds are added to the center and old seeds are pushed out following a set of rules. Under this dynamic method, the angles and spirals are an emergent aspect of the system, rather than the explicit result. This website describes how such a dynamical system could be modeled.

Although the Fathom/TP model does not model the dynamical system that underlies phyllotaxis, it’s fun to play with in its own right. You can manually alter the base_angle attribute as suggested by the GSP article. If you add a parameter (a slider) to help you vary the angle, you can obtain a whole family of spiral/ray patterns whose properties you could take a closer look at. Different combinations of angles and sliders will give you various levels of control over the image.

For example, change the formula for base_angle to base_angle = pi*(1+sqrt(5))*base, and create a slider “base”. The image below shows the spirals obtained for base = 1…6.

Update: Here is an example of how to draw spirals like this in R.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to dan.mackinnon*