About Mathematics Commentary
Chapter 1. Getting Started
What this chapter is aboutThis chapter has an important role to play. It is designed to give students experience with unfamiliar mathematics and to
take them quickly to an example of a fractal, and thus to an area of active 21st century mathematical research. The activities
may indeed be trivial and peripheral to that study but it should at least give the students, if you stress this with care, a
sense of accomplishment beyond that of their science student friends.
The chapter also introduces students to the panels that will carry much of the calculating weight of the course.
For straightforward calculation they can use their own calculators or the apps on their iPhones or iPads.
Section Notes and ActivitiesSection 1.1. Nowadays our students are often more familiar with technology than we are, and the barcode squares next to each panel
may serve as a perfect example. If using these barcodes is completely new to you, here is what you will need to do.
(Follow these instructions closely because you will wish to share the information with any students unfamiliar with them as well.)
1. With a smart phone or pad, download one of the free QR Readers from the App Store.
2. Open the QR Reader app.
3. Direct your device’s camera at one of the text panel barcodes. Your device screen will help you find the square.
You will quickly get used to this.
That’s it. The barcode will transfer you directly to the panel. (Occasionally, your camera may misread the code and
send you to a completely unrelated source. In that case, simply try again.)
There is, of course, an alternate means to access the panels. You can directly access am.sienacs.com and choose the
chapter and panel from the menu. Clearly the approach using the barcode is far more efficient, but you may need to use this
method if you are working at your computer.
In your class when you introduce your students to this technique, you might then address exercise (1.1.2), using Panel 1.1.1. While
you might be tempted to develop an algorithm to show how Panel 1.1.1 works here, we strongly recommend that
you hold off doing so until late in Chapter 2, which is about algorithms. Just let the students have fun with this panel
for now. Also projecting these images of hexagons and octagons on a screen, can add to your discussion of the optical illusions that
hexagons and octagons present.
We also recommend that you work on exercise (1.1.4) as a class activity, collecting data from Panel 1.1.2, and seeking various
ways (not necessarily algebraic) to represent the relationship between the number of loops of bricks around the central brick and
the total number of bricks.
Section 1.2. reminds students about order of operations and gives them some experience calculating with a scientific calculator. To assure
yourself that your students are using their calculators correctly, you might want to go through many of the exercises with them.
You might ask about exercise (1.2.5): The definition of a tower of powers says that you should evaluate from the top down. Does
that matter? In other words, is top down the same as bottom up?
Section 1.3. contrasts situations in which disorder enters into what seem like straightforward calculations (rounding calculations and
repeating computer simulations) with situations in which seeming disorder produces order (triangle midpoint locations). The
Sierpinski and dragon fractals are then introduced as belonging in a kind of Never-Never Land between the two. (It is important
to point out, however, that technically all fractals belong in the second group, although the order becomes increasingly complex
as the process continues.)
While fractals play a small role in this discussion of order and disorder, the subject can prove very attractive to your students.
Because you may wish to take advantage of this interest, we offer some suggestions about resources here.
Dragon Curve and Fractals. When you run Panel 1.3.4, the Dragon fractal is generated one segment at a time. Treated as a city walk as in Panel 1.3.3,
the route turns are: RRLRRLLRRRLLRRLLLRLRRRLLR…, or in terms of Right, Left, Up and Down as: RDLDLULDLURULURULDLDLURURUR….
(These can be checked by running Panel 1.3.4 and following the early stages of the route.) While there appears to be no pattern,
there is one but it is complicated. The complex math that drives this pattern is offered on the Wikipedia site,,
en.wikipedia.org/wiki/Dragon_curve#Code, which also demonstrates some alternate ways of drawing the curve. A way to draw at least
the early stages of the curve (on centimeter graph paper) is at:
bentrubewriter.com/2012/04/25/fractals-you-can-draw-the-dragon-curve-or-the-jurassic-fractal/.
You or one of your students might want to show how some of the early stages of the pattern can be accomplished by folding a strip of paper.
(See m.youtube.com/watch?v=wCyC-K_PnRY or the Wikipedia article section titled “[Un]folding the Dragon.”)
The Koch snowflake is a simpler example of a fractal that is also easy to illustrate. For more on this curve, see
en.wikipedia.org/wiki/Koch_snowflake or drawing instructions at: bentrubewriter.com/2012/04/24/fractals-you-can-draw-the-koch-snowflake/.
A remarkable feature of this curve is that, at each stage the perimeter of the curve is increased. Thus, if continued without limit,
the figure results in a finite area within an infinitely long border. You may be tempted to show some of the mathematics involved as
the perimeter and area increase; we recommend that you do not do this, however, because for many of these students you will simply
confirm their suspicion that you are going to overwhelm them with math technique. At this stage you might better simply show the first
two stages of the Koch snowflake calculator program that produces the curve and ask, “Are the area and perimeter increasing?”
The answers are straightforward. Some of your students may be attracted to the design aspects of fractals and, especially when color
is added, their near-hypnotic quality can be interesting. The Wikipedia site already referenced has some examples (see “Tiling” and
the sections that follow) as do: www.eschertile.com/pic/dragon3.gif, and stateoftheartsnyc.wordpress.com/2018/07/02/dragon-curve/
To extend your own mathematical background see the excellent essay, “Dragon Curves Revisited” by S. Tabachnikov at:
www.personal.psu.edu/sot2/prints/DragonCurves.pdf. For the more general subject of fractals, see: en.wikipedia.org/wiki/Fractal,
and its references.
Once again, we note that the point of this section is to expose your students to a mathematical topic that is currently being
actively studied and that is very likely unfamiliar to their STEM student colleagues. If you make a point of this, it should give
your students some satisfaction and you may wish to discuss further aspects of fractals, beware of retreating into lecture mode.