Image of the Week #144

A newton monster fractal with internal organs!



Adding up infinitely many numbers

topThis week’s post looks at circumstances when you can add up an infinite number of numbers and get a single number as the answer. The classical example is adding one-half plus one-fourth plus one-eighth and so on, forever. The picture at the left shows that the total is (i) bigger than any number smaller than one and (ii) no more than one. This means that the total is, in fact, one. This follows from the fact that the set of numbers bigger than any number smaller than one but not bigger than one is… …one. When Occupy Math teaches this in calculus class, he uses a technical trick called a limit, but the reasoning that says the total is one is solid as presented.

The trick for the picture at the beginning of the post is that, every time we add one of the numbers to the picture, the part that’s left gets half as big. If you divide something in half forever, it becomes smaller than any positive number — which means it becomes zero. This means that the total of one-half plus one-fourth and so on is within zero of one; in other words, it is one.

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Morphing Shapes, Morphing Rules

topThe shapes above are all circular rings, which may seem sort of odd since one is a diamond, one looks like a circle, and one is almost square. There is a mathematical trick that makes these all circles: the way distance is measured is different for each of these. The normal circular ring is based on squares and square roots, which gives us what we think of as the “normal” distance measure. The diamond is based on first powers and the almost square ring is based on fourth powers. Cool, no?

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Asymmetric Economic Power

topOccupy Math had occasion to travel to Kingston, Ontario a few weeks ago. Since winter in Canada is a chancy time to drive on the highways, he took the train. The walk to the train station was along Yonge Street in Toronto (shown at the top of the post), a major shopping district that Occupy Math visits several times a year. The walk was also before dawn and there were relatively few people about. This made it easy to see that most of the interesting single-proprietor shops had been replaced by generic chain stores. Why that is happening is the topic of today’s post.

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Can you add it up? An activity.

This post is the next of Occupy Math’s series on problem factories. Problem factories are a body of mathematical knowledge that, once you understand it, lets you generate many problems and, hopefully, multiple types of problems. The class of puzzles in this post is exemplified by the question “Can you write 25 as the sum of consecutive numbers?” A really clever student might realize that the number 25, by itself, is one consecutive number but if we forbid this answer by fiat, then there is a picture that answers the question. How far can this sort of picture-proof be pushed?


The picture gives us 3+4+5+6+7=25, but there is also 12+13 and 25 by itself. There are also some answers that use negative numbers like -2-1+0+1+2+3+4+5+6+7=25. The fact there are often many answers makes this problem more interesting.

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Grading with rubrics, why and why not.

topThe Eberly Center at Carnegie Mellon University says:

A rubric is a scoring tool that explicitly represents the performance expectations for an assignment or piece of work. A rubric divides the assigned work into component parts and provides clear descriptions of the characteristics of the work associated with each component, at varying levels of mastery.

Occupy Math is sometimes one of the three instructors for a course called Topics in Bioinformatics. This is a course where new students in the bioinformatics program read, discuss, and evaluate papers that introduce the research areas in bioinformatics. The original grading rubric for this bioinformatics course, if followed carefully, would have flunked every student that chose a mathematical or computational tool paper to evaluate. This problem arose from cultural disjunctions between different disciplines in an interdisciplinary field — but it shows that rubrics are potentially dangerous. This post critically examines rubrics which can be useful or deadly.

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