## Girls’ Angle Bulletin, Volume 8, Number 3

Volume 8, Number 3 of the Bulletin kicks off with the first part of a 2-part interview with Hamilton College Assistant Professor of Mathematics Courtney Gibbons.  In this first half, she discusses how she got interested in math, tells us about how she learns, studies, and creates math, and explains the notions of “fixing set” and “fixing number.”  She also tells us about some of her goals as a mathematician.

The best way to learn math is to do lots of problems: after all, that’s what math is! – Courtney Gibbons

## We’ve Got Snowball Problems

Current weather in New England makes it hard to not think about snow. So here are two snowball inspired math problems. We welcome your solutions (send to girlsangle “at” gmail.com). Perhaps we’ll post solutions when the last traces of snow have cleared from Boston streets.

Both problems involve a snowball in the shape of a perfect sphere. Mark a point on the surface of the snowball. The snowball rolls without slipping. Also, there is no unnecessary rotation. (In other words, the snowball rotates only around an axis perpendicular to the plane of the great circle that is tangent to the path that the snowball travels.) In each problem, the snowball starts with the marked point touching the surface it is rolling on.

Snowball Problem 1. If the snowball rolls about on a plane, what is the locus of all the points in space that the marked point can visit?

Snowball Problem 2. If the snowball rolls about on another sphere of the same radius, what is the locus of all the points in space that the marked point can visit?

## Girls’ Angle Bulletin, Volume 8, Number 2

After reading Stained Glass Angles in Volume 7, Number 4, Marion Walter happened upon an amazing stained glass window by John Rose at the Eugene Public Library in Oregon. That inspired her to suggest a Math Buffet column featuring mathematically inspired stained glass windows. One thing led to another, and in this issue, we feature stained glass windows from 7 designers. The cover shows a detail of a stained glass window designed by Millie Wert, a graduate of Harpeth Hall.  (Added January 27, 2015: For more math inspired stained-glass windows by students of Thaddeus Wert at Harpet Hall, please visit his blog.) Continue reading

## Go Digital! A Crossword Puzzle Raffle

(Added January 7, 2015: This raffle is now closed. Thank you to all who entered. Congratulations to F. Dangerfield of Massachusetts who won the member draw and to B. Jackson of Wisconsin who won the general draw!)

At Girls’ Angle, we do love digits!

Solve this crossword alone or with friends and relatives.  Scan in or take a digital photo of your completed grid and send it to girlsanglepuzzler “at” gmail.com. We will randomly draw from the correct answers received by midnight on January 4, 2015 to select a “winner” and send the winner a small prize. Girls’ Angle members will be put into a separate pool for a different prize.

This crossword puzzle is dedicated to Connie Chow.

Across

1. Rest
6. Fall mo.
9. Sub
14. Breathing
15. Greek letter
17. Hushed
18. Goal
19. Vermeer product
20. Sunrises and hatchings
23. Soft cheese
24. It often gets beat
25. Rights wrongs
27. City in 54D
31. Donor
33. Unlike Smith
34. Acts like a ghost
39. Für Elise form
41. Type of health plan: Abbr.
42. Rise over run
44. Increases
45. Forest and jungle
48. Divisible by 2
49. Catches up to
50. Most frigid
52. Triangular sail, or, a 15-year-old who lives near the Endeavour?
55. Hamburg’s commission
56. Drive the getaway car
57. Equal areas in equal times
64. Burdens
66. Overly
67. Baking bean
69. Trig function
70. Seaweeds
71. Veracity
72. Time unit’s plural form: Abbr.
73. Name with a double consonant

Down

1. Docile
3. Flying diamond
4. Follows for or what
5. Pentagon property
6. Type of ape
7. Elegant
8. The Princeton Companion to Mathematics, for example
9. Hawaii is the last of these: Abbr.
10. Samson’s bane?
11. No longer useful
12. Corrosive
13. Kids
21. They can be bruised
22. Oft used file command
26. Place to eat
27. Brown
28. Continuous image of the circle
29. Writes
30. Total
31. Most exciting match
32. Fe
35. Soften in the mouth
36. Zero to Federer
37. What cosine does to sine
38. Not straight
40. Shrek
46. File’s partner
47. Tails
49. Loathe
51. Someone you only drive to work with?
52. Fastener
53. Haliotis
54. State of 27A
55. Clean with string
58. Draw
59. Low quality
60. Shelf with a view
61. Plastic brick
62. Not home
63. Gave more importance
65. Short

Good luck!

(We will not use your contact information for any purpose other than to deliver your prize, should you win. After the winner has been selected, all emails received will be promptly deleted. At the winner’s discretion, we will let you know who won. Anyone who makes more than one submission will be disqualified! Sorry! Also, this offer is only valid in those states in the United States where such things are legal. There is no fee to enter this puzzle contest.)

## Girls’ Angle Bulletin, Volume 8, Number 1

This time, we’re leaving it up to readers to guess what our latest cover represents.

This past summer, Girls’ Angle program assistant Margo Dawes traveled to New York City to interview Cathleen Morawetz, Professor Emerita at the Courant Institute of Mathematical Science at New York University.

Last year, Professor Jennifer Roberts, the Elizabeth Cary Agassiz Professor of the Humanities at Harvard University, wrote an essay for Harvard Magazine about “creating opportunities for students to engage in deceleration, patience, and immersive attention.” Inspired by that essay, and recognizing that this could be a valuable exercise in mathematics as well, we attempt to give you such opportunities in mathematics with our new column, Meditate to the Math.  Our first installment features the 9-point circle. Instead of reading about the 9-point circle, we encourage readers to find a comfortable, quiet place, and contemplate a geometric figure. We hope this will be a way to take part in the process of mathematical discovery.

Next, follow Emily and Jasmine as they contemplate 5-pointed stars. If any of our members or subscribers have an exciting story of mathematical discovery of their own, we welcome you to tell us about it!

This issue’s Learn by Doing addresses quadratic residues. Last Summer’s batch of Summer Fun Problem sets included one on quadratic reciprocity by Cailan Li.  But before quadratic reciprocity, there are lots of things to say about quadratic residues.  We explore some of those neat properties here.

Last issue, Anna made a neat discovery about stereographic projection and paraboloids of revolution. As often happens with mathematical theorems, the first proof is messy and then spiffier proofs are found later. In this issue’s Anna’s Math Journal, Anna finds a much nicer proof and then applies the result to describe a few more observations about paraboloids of revolution.

While contemplating paraboloids of revolution, Anna also came upon a way to understand the radical axis of two circles. This observation seemed more convenient to write an article on because she came to this understanding without writing anything down. She explains in Seeing the Radical Axis. Lightning Factorial supplements her article by briefly defining the radical axis for readers not yet familiar with the concept.

A few weeks ago at the Girls’ Angle club, some members helped to simplify Lunga Lee’s excessively long descriptions of various functions. You can try your hand at this in Function Madness.

Also inside are another installment of Math In Your World, some exercises about real algebraic varieties (to follow-up on Dr. Zamaere’s introduction of them in her interview in the previous issue), and some notes from the club, which include a summary of Emily Pittore’s recent visit.  Emily is a robotic vision engineer from iRobot, the maker of the Roomba vacuum cleaner.

We hope you enjoy it!

Finally, a reminder: when you subscribe to the Girls’ Angle Bulletin, you’re not just getting a subscription to a magazine. You are also gaining access to the Girls’ Angle mentors.  We urge all subscribers and members to write us with your math questions or anything else in the Bulletin or having to do with mathematics in general. We will respond. We want you to get active and do mathematics. Parts of the Bulletin are written to induce you to wonder and respond with more questions. Don’t let those questions fade away and become forgotten. Send them to us!

We continue to encourage people to subscribe to our print version, so we have removed some content from the electronic version.  Subscriptions are a great way to support Girls’ Angle while getting something concrete back in return.  We hope you subscribe!

## Multiplication Sculptures and Sums of Cubes

A 15 by 15 multiplication sculpture built by Jane Kostick.

Take a multiplication table and build a column of cubes over each entry. For each column, use as many cubes as the product it sits upon. The result is a “multiplication sculpture” or “multiplication tower.” The picture shows a 15 by 15 multiplication sculpture built by Jane Kostick in 2008. For more examples, check out Maria Droujkova’s photo collection at moebius noodles.

Last spring, club members at Girls’ Angle built a 4 by 4 version out of cubes that were 3 inches on a side. They thought about its properties, such as how many cubes make up an N by N multiplication sculpture.

We’re going to address this last question and end up with a nice, concrete proof of a famous algebraic identity.

Over the xy entry, there are xy cubes, so we have to add up all products xy where x and y range over the values from 1 to N, and this totals

$(1+2+3+\dots+N)^2 = (\frac{N(N+1)}{2})^2$.

Wait! Where else does that expression occur?

It is also the sum of the first N (positive) perfect cubes! In math notation,

$1^3 + 2^3 + 3^3 + \dots + N^3 = (\frac{N(N+1)}{2})^2$.

The implication is that the number of cubes sitting over the last row and column of the multiplication table must be $N^3$. If we can show this directly, we’d have a nice, concrete proof of the sum of cubes formula.

The columns of cubes over the last row of the multiplication table form a staircase with steps of height N. Saw these N columns off and get a flat, staircase-shaped plank, N cubes long and $N^2$ cubes high. Next, saw off the columns of cubes over the last column (of the multiplication table). You’ll get an almost identical staircase-shaped plank. The only difference is that the last step of total height $N^2$ is gone because it was removed when the columns over the last row (of the multiplication table) were sawed off.

Turn one of these planks over, and the two planks will fit together perfectly to form an $N$ by $N^2$ rectangle, and an $N$ by $N^2$ rectangle has $N^3$ cubes in it!

Thus, a secret key to the identity

$1^3+2^3+3^3+\dots+N^3 = (\frac{N(N+1)}{2})^2$

is hidden in plain sight in the multiplication table that many of us learned in elementary school!

To read about how Jane made this 15 by 15 multiplication sculpture, and see hints about more of its properties, check out the December, 2008 issue of the Girls’ Angle Bulletin, pages 12-14 and 25-28.