3D printing of visual mathematics

Mathematicians have used visual representations of abstract mathematics for many years. With the recent availability of inexpensive 3D printers, it’s now easier to build these objects. Prof. Rida Mirie has started to develop an expertise in this area. Using a DaVinci printer, he is working on printing objects that match the surfaces that students encounter in courses such as Calculus III.

This has come just at the right time for Prof. Tibor Beke, who is teaching a section of Explorations in Math to students in the Humanities, Fine Arts and Social Sciences who have accepted the challenge to explore some mathematics as a somewhat higher level than is normally offered to students on our South Campus.
We all know the formula \[ 1+ 2 +3+ \dots + n = \frac{n(n+1)}{2} \]
that can be justified in a number of ways — by induction, but adding the left-hand sum to itself in the reverse order, or by decomposing an \(n\) by \(n+1\) rectangle into two congruent pieces, each of whichcontains \(1+2+3+\dots+n\) unit squares. But what about\(1^2 + 2^2 + 3^3 + \dots + n^2 \)? A nice way to visualize such a sum is as the number of cubes in a skewed “Mayan pyramid.” Here is are six Mayan pyramids printed by Rida that are a visual representation of \(1^2 + 2^2 + 3^2 + 4^2 \).
six mayan pyramids
Three such pyramids can be combined to form a cuboid with a set of steps next to one of the faces. The steps in two such formations can, if you orient the pieces correctly, be fit together.
Thumbnail image for maya_2.jpg
When this is done, you get a single cuboid. In this case, it’s a \(4 \times 5 \times 9\) cuboid, demonstrating that \(6(1^2 + 2^2 + 3^2 + 4^2) = 4 \times 5 \times 9 \). The 5 in this equality is one more the 4 and 9 is one more than \(2 \times 4 \).
Thumbnail image for Thumbnail image for maya_3.jpg
This configuration works for the sum of the first \(n\) squares for all positive values of \(n\), which demonstrates a general identity, after dividing by 6: \[ \sum _{k=1}^{n } {k^2} = \frac{n(n+1)(2n+1)}{6} \]
The nice thing about having a tactile representation of this fact is that students can actually put the pieces together and see how it is really not dependent on the number of squares. “Proofs with no words” such as this one have traditionally been accepted as valid proofs. They are limited to our three dimensions, but the printing of complex objects opens up possibilities that we haven’t had until now.

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Ken’s Khronicles – October 2014

This is the first Khronicles since last December, and we’ve been busy with several personnel changes since then.

New Hires
Early in January of this year we were authorized to do faculty searches for two tenure-track positions, a statistician and an applied mathematician. Although it was a late start, we are happy to announce that both searches were successful.
Dr. Jong Soo Lee, a statistician who was most recently at the University of Delaware, has joined us this fall. His general research areas are functional data analysis, nonparametric statistics and the application of statistics. His Ph. D. was earned at Rice University with a thesis titled Aspects of Functional Data Inference and Its Applications (Advisor: Dennis Cox).
Dr. Hung Phan, an applied mathematician who was most recently at the University of British Columbia, Okanagan, will join us in the spring. His general research areas are Optimization, Numerical Methods, and Variational Analysis. His Ph. D. was earned at Wayne State University, with a thesis titled New Variational Principles with Applications in Optimization Theory and Algorithms (Advisor: Boris Mordukhovich).
This infusion of new personnel comes just in time to offset losses due to retirement. Charlie Byrne retired after being with us since 1986. He served as department chair for a term, and graduate coordinator for many years. His expertise in areas such as optimization and image processing is hard to replace.
Congratulations to Charlie on his recently published book, An Introduction to Optimization.


In addition, we’ve lost the services of Alan Kaplan, who retired after forty years of service. His many contributions to the department will be missed.
More Hiring

More good news is that we’ve been authorized to do two more searches for tenure-track faculty. We’ve decided to search for another statistician and a mathematician. The research area for the mathematician is a bit less focused that in our earlier search. To see details of our postings, go to the UML Jobs site.
More News
There is plenty of other news to report, but I intend to make these postings more frequent. So I’ll close here.

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Market Basket spending/non-spending, and more

I guess the biggest news centered around Lowell this summer has been the Market Basket fiasco. A couple of weeks into the boycott, I thought I’d create a visual to contrast our spending at different markets in the past year. Naturally, I used Mathematica. Here is a version of the plot I posted on a Facebook group site related to Lowell:

MRKT BSKTSHAWspending.jpg
Naturally, the plot labeled MRKT BSKT is our spending at Market Basket. The SHAW graph is actually a combination of our spending at Shaw’s and Hannaford’s. I think I stopped at Shaw’s around February and bought a few items, but other than that, spending there started at the same time as the boycott.
Instead of making this a one-time exercise, I decided to develop a function that would create other graphs like this one. My main summer project has been to rebuild the stairs at my son’s house in Chelmsford, adding a small deck. Out of convenience, my main spending on the project, which started in mid-July, has been at three locations. They are my local NH Home Depot (no sales tax!), where I’ve bought most of the smaller items I could fit into my Subaru, the Lowell Lowe’s, which is five minute from the project and where I’ve bought several last minute items I didn’t expect to need, and finally, Friend Lumber from whom I’ve had lumber delivered. Here is the spending plot at those locations for 2014, with another ~$100 spending to go at this point:
LOWEHOME Dspending.jpg
One more example, here is my coffee spending – this is flawed in that it is only the debit card spending, so a lot of my purchases at Einstein’s during the year are not included. A&E is my local coffee roaster, and I definitely spend more there that anywhere else, even if I took into account cash sales. Also, I use the Starbucks app on my phone, which is why larger, less frequent purchases are recorded there.
My spending on coffee
If you’d like to try this with your spending, you can download a copy of the Mathematica Package that creates these graphs. The package assumes you have a Bank of America debit card. You would download activity from your account and use the file path to that .csv file on your computer to specify the data file. With a bit of Mathematica expertise, it shouldn’t be hard to adapt the package to other banking systems.

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Math Majors stand out at the Uteach Conference in Austin

The 2014 UTeach conference at the University of Texas Ausin brought together faculty, students and staff from 40 universities with UTeach programs and five universities aspiring to build a UTeach program. The UMass Lowell UTeach team included two UML math majors, Erinn McLaughlin and John Romano.

ErinnMcLaughlin, togetherwithMichelle Scribner-MacLean(Graduate College of Education) and William Morton (Lowell NationalHistoricalPark), described how she and her classmates teamed with the National Park to create science and math projects for high school students. The work was part of the requiredUTeachcourse, Project Based Instruction.


At a poster session that included dozens of student submissions, John Romano won the best student poster for a research project for the work he did as an intern with M2D2 and Lowell High School. Erinn also displayed a poster on the mathematics of water wheels.


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Curriculum Guides in Mathematics: JMM2014 update

I attended a panel discussion on curriculum guides at the 2014 Joint Mathematics Meetings last week. Here are a few comments on them.

  • The CUPM Curriculum Guide is produced by the MAACommittee on the Undergraduate Program in Mathematicsto guide mathematics departments in designing curricula for their undergraduate students.The 2004 version was the last to come out. The new version will be out in 2015. We were told that a draft will appear atmaa.org/cupm in the near future.
  • In 2012, the Conference Board of the Mathematical Sciences issued the latest recommendations for teacher preparation in mathematics: The Mathematical Education of Teachers II (MET2). A few highlights of the new recommendations:
    1. Elementary teachers should take four mathematics courses on elementary school mathematics. This doesn’t mean that the mathematics they are taught are elementary. The objective is to give teachers a deeper understanding of the mathematics that is taught in elementary grades. For example, while an elementary school teacher may teach division, coursework might include continued fractions or a study of the periodic nature of decimal fractions.
    2. Recommendations for middle school teachers include at least 24 credits of mathematics, including at least 15 credits designed specifically for future middle grades teachers that address essential ideas in the middle school curriculum.
    3. It is still recommended that prospective High school teachers complete coursework equivalent to that of a mathematics major. One change is that at least nine credits involve advanced study of secondary mathematics.
  • TheAmerican Statistical Association (ASA) will be releasingThe Statistical Education of Teachers (SET)in 2014. It is expected to put a greater emphasis on data analysis.
I think that a few developments at UMass Lowell have put us in a good position with respect to these recommendations. A few years ago, the College of Education and Mathematical Sciences Department collaborated with other UMass campuses on the development of mathematics courses for prospective elementary school teachers. This gives us a good start toward being in line with recommendations at that level.
UTeach UMass Lowell helps us at the middle and high school levels. Functions and Modeling (92.210), which is required for mathematics certification, revisits many high school topics from an advanced point of view. Research Method (UTL.302), which is required of all UTeach students, is a data analysis course that matches both MET2 and SET recommendations. Finally, the inquiry-based approach that many UTeach courses emphasize is consistent with that of all three curriculum guides.
There will be more for us to do to address these recommendations, but I think we are on the right track!

Ken’s Khronicles – December 2013

I hadn’t expected to be writing this column up until last May, but here I am serving as chair of Mathematical Sciences again after a 14 year break. I had been chair for nine years, roughly coinciding with the Clinton administration. There are several news items to report, but first I’d like to reflect on what’s changed in 14 years.
  • The last time I was chair we were housed in Olsen, now we’re in Olney. With new buildings starting to pop up on campus, several departments, including us, could be moving again in the coming years.
  • Since my first ‘retirement’ as chair, we’ve seen the retirements of Professors Yin, Makovoz, Weinberg, Spiegel, Mueller, Winslow, Berkovits, Ruskai, and Samarov. We currently have seven tenured faculty and seven Instructors who have joined us since then.
  • When I left the chair’s position in the 20th century, we offered a two semester version of Calculus I, Calculus IA and Calculus IB. After trying ‘Preparation for Calculus’ for a few years, we now we have a new two semester version of Calculus I. Same name, but different catalog numbers. Everything is cyclic.
  • Scholarships: Toward the end of my first run as chair, Russ and Mary Bedell’s generosity had just brought us our first endowed scholarship. As you can see below, we’ve been fortunate to have two more endowed scholarships started since then and hope to be able to announce more in the future.
  • Finally, Tangents didn’t exist in 1999 – the first issue was in 2002. So now I will move to the present.
One of the main reasons why I agreed to take on the chair’s position is that Kiwi has agreed to help me out as associate chair. In addition, I’m getting lots of help from other members of the faculty and, of course, Cori Lee, our Administrative Assistant. I thank all of them for helping make my return reasonably smooth.
It’s helped that I’ve been able to divest some of the things I was doing. Steve Pennell has started his ‘chair’s retirement’ by taking over as coordinator of the Industrial Math PSM program. Also, I had been editing Tangents since 2002 and I’m glad to announce that Jim Propp has agreed to take over as editor.
Tibor Beke has returned to us after a year on sabbatical traveling through Europe and the States in 2012-13 and Ravi Montenegro has left us for a 2013-14 sabbatical to collaborate with researchers in Japan and New Zealand.
Math majors Olivia Demers and Mary Mersereau had been working with Shelley Rasmussen in doing research on the mathematics of weaving. They recently shared their enthusiasm for the subject with some children in a local summer camp.
In the past year, Industrial Mathematics PSM Students Isaac Duodu (Putnam Investments) and Lauren Edwards (Genscape) completed internships that gave them valuable experience to further their careers. Undergraduates Gifty Bado and Tyler Gilzinger spent their summers as Co-Op students at Putnam Investments. After completing a co-op position at Mercury Computer, recent B. S. graduate Owen Welch accepted a full-time position at Mercury. Krithika Manohar, who graduated last May, is in her first semester of the Applied Math doctoral program at the University of Washington.
Last May, we held our annual Awards Ceremony and Alumni Reception where we presented the following awards:
  • Outstanding Graduate Student: Nour Almansour
  • Shapiro Scholarship: Chris Leger
  • Bedell Scholarship: Tyler Gilzinger
  • Zamanakos Scholarships: David Campbell, Mary Mersereau and Kevin Southwick
  • Hall Prize: Kevin Cerritelli.
The 2014 awards ceremony will be held at the UML Inn and Conference Center on April 25 from 5:30 to 7:30. Alumni are always welcome to attend. It’s a great setting.

Visualizing the 2013 Schedule using RAW

RAW is described as “The missing link between spreadsheets and vector graphics. It’s a really easy way to generate interesting data visualizations from spreadsheet data. For more information: http://raw.densitydesign.org

Here is a visualization of the Fall 2013 Math Schedule showing the times at which each course is offered.

Elem.Math for Tchng:Num.&Oper.MoWe 11:00AM – 12:15PMQuantitative ReasoningMoWeFr 8:00AM – 8:50AMMoWeFr 9:00AM – 9:50AMMoWeFr 10:00AM – 10:50AMMoWeFr 11:00AM – 11:50AMMoWeFr 12:00PM – 12:50PMTuTh 8:00AM – 9:15AMTuTh 9:30AM – 10:45AMTuTh 12:30PM – 1:45PMTuTh 2:00PM – 3:15PMSI for Quan. Reasoning & StatsTuTh 9:30AM – 10:20AMTuTh 11:00AM – 11:50AMTuTh 12:30PM – 1:20PMPrecalculus Mathematics ITh 6:30PM – 9:20PMTBAManagement PrecalculusMoWeFr 8:00AM – 8:50AMMoWeFr 9:00AM – 9:50AMMoWeFr 10:00AM – 10:50AMMoWeFr 11:00AM – 11:50AMMoWeFr 12:00PM – 12:50PMMoWeFr 1:00PM – 1:50PMMoWeFr 2:00PM – 2:50PMMoWeFr 3:00PM – 3:50PMTuTh 9:30AM – 10:45AMTuTh 11:00AM – 12:15PMTuTh 2:00PM – 3:15PMTuTh 3:30PM – 4:45PMManagement Pre-Calculus SIFr 9:00AM – 9:50AMFr 11:00AM – 11:50AMFr 12:00PM – 12:50PMFr 1:00PM – 1:50PMFr 2:00PM – 2:50PMTh 8:00AM – 8:50AMTh 9:30AM – 10:20AMTh 11:00AM – 11:50AMManagement CalculusTu 6:30PM – 9:20PMMoWeFr 9:00AM – 9:50AMMoWeFr 10:00AM – 10:50AMMoWeFr 1:00PM – 1:50PMMoWeFr 2:00PM – 2:50PMTuTh 2:00PM – 3:15PMTuTh 3:30PM – 4:45PMManagement Calculus SIFr 9:00AM – 9:50AMFr 11:00AM – 11:50AMFr 12:00PM – 12:50PMFr 1:00PM – 1:50PMTh 8:00AM – 8:50AMTh 9:30AM – 10:20AMTh 11:00AM – 11:50AMPrecalculus Math IIWe 6:30PM – 9:20PMCalculus AMo 6:30PM – 9:20PMCalculus BMo 6:30PM – 9:20PMCalculus IAMoWeThFr 8:00AM – 8:50AMMoWeThFr 9:00AM – 9:50AMMoWeThFr 10:00AM – 10:50AMMoWeThFr 11:00AM – 11:50AMMoWeThFr 12:00PM – 12:50PMMoWeThFr 1:00PM – 1:50PMMoWeThFr 2:00PM – 2:50PMMoWeThFr 3:00PM – 3:50PMMoTuWeFr 11:00AM – 11:50AMMoTuWeFr 12:00PM – 12:50PMCalculus IBMoTuWeFr 10:00AM – 10:50AMCalculus IMoTuWeFr 8:00AM – 8:50AMMoWeThFr 8:00AM – 8:50AMMoWeThFr 9:00AM – 9:50AMMoTuWeFr 10:00AM – 10:50AMMoTuWeFr 11:00AM – 11:50AMMoTuWeFr 12:00PM – 12:50PMMoTuWeFr 1:00PM – 1:50PMMoWeThFr 1:00PM – 1:50PMMoWeThFr 2:00PM – 2:50PMCalculus IIMoTuWeFr 10:00AM – 10:50AMMoTuWeFr 11:00AM – 11:50AMMoTuWeFr 1:00PM – 1:50PMMoTuWeFr 3:00PM – 3:50PMMoTuWeFr 4:00PM – 4:50PMMoWeThFr 8:00AM – 8:50AMMoWeThFr 9:00AM – 9:50AMMoWeThFr 12:00PM – 12:50PMMoWeThFr 2:00PM – 2:50PMCalc. for the Life Sciences IIMoTuWeFr 11:00AM – 11:50AMHonors Calculus IMoWeThFr 11:00AM – 11:50AMFunctions and ModelingTuTh 9:30AM – 10:45AMLinear Algebra IWe 6:30PM – 9:20PMMoWeFr 9:00AM – 9:50AMMoWeFr 8:00AM – 8:50AMCalculus CWe 6:30PM – 9:20PMCalculus IIIMoTuWeFr 8:00AM – 8:50AMMoTuWeFr 9:00AM – 9:50AMMoTuWeFr 10:00AM – 10:50AMMoTuWeFr 12:00PM – 12:50PMMoTuWeFr 1:00PM – 1:50PMMoTuWeFr 2:00PM – 2:50PMMoWeThFr 10:00AM – 10:50AMMoWeThFr 11:00AM – 11:50AMMoWeThFr 12:00PM – 12:50PMMoWeThFr 1:00PM – 1:50PMDifferential EquationsMo 6:30PM – 9:20PMMoWeFr 9:00AM – 9:50AMMoWeFr 11:00AM – 11:50AMEng Differential EquationsMoWeFr 1:00PM – 1:50PMMoWeFr 2:00PM – 2:50PMMoWeFr 10:00AM – 10:50AMMoWeFr 12:00PM – 12:50PMHonors Calculus IIIMoWeThFr 12:00PM – 12:50PMIntroduction to StatisticsMo 6:30PM – 9:20PMTu 6:30PM – 9:20PMTBAMoWeFr 9:00AM – 9:50AMMoWeFr 10:00AM – 10:50AMMoWeFr 11:00AM – 11:50AMMoWeFr 12:00PM – 12:50PMMoWeFr 2:00PM – 2:50PMTuTh 8:00AM – 9:15AMTuTh 9:30AM – 10:45AMTu 9:30AM – 10:45AMTuTh 11:00AM – 12:15PMTu 12:30PM – 1:45PMTuTh 12:30PM – 1:45PMTuTh 2:00PM – 3:15PMTuTh 3:30PM – 4:45PMIntro to Applied Math ITu 6:30PM – 9:20PMMoWeFr 11:00AM – 11:50AMDiscrete Structures IWe 6:30PM – 9:20PMTBAMoWeFr 9:00AM – 9:50AMMoWeFr 12:00PM – 12:50PMDiscrete Structures IIMoWeFr 10:00AM – 10:50AMTuTh 12:30PM – 1:45PMSymbolic LogicTuTh 3:30PM – 4:45PMIntro to Data AnalysisTuTh 11:00AM – 12:15PMSenior Seminar ITBAApplied StatisticsTuTh 3:30PM – 4:45PMProb & Stats IWe 6:30PM – 9:20PMTuTh 2:00PM – 3:15PMMathematical AnalysisMoWeFr 8:00AM – 8:50AMProb & Math Stat ITuTh 2:00PM – 3:15PMAbstract Algebra ITh 6:30PM – 9:20PMGeometryMo 3:30PM – 6:15PMSenior Seminar IITBASenior Seminar IIITBASelected TopicsTBAReal AnalysisMo 6:30PM – 9:20PMProbabilty & Math StatsTu 6:30PM – 9:20PMApplied Mathematics ITu 6:30PM – 9:20PMApplied Math for Life Scien.Mo 3:30PM – 6:20PMComputational MathematicsTh 6:30PM – 9:20PMOptimizationWe 6:30PM – 9:20PMMultivariate StatisticsWe 6:30PM – 9:20PMExperimental DesignTh 6:30PM – 9:20PMSelected Topics in MathematicsTBAThesis Review (Thesis Research)TBA

Mathematical Weaving

In recent years, Prof. Shelley Rasmussen has be doing research into the mathematics of weaving. She has gotten several students involved in her work. Most recently, math majors Olivia Demers and Mary Mersereau had been working with Shelley. They recently shared their enthusiasm for the subject with some children in a local summer camp. For more information on the mathematics of weaving: Shelley_Rasmussen@uml.edu.


Embedded Sage Cells


sagecell.makeSagecell({“inputLocation”: “.sage”});
$(function () {
// Make the div with id ‘mycell’ a Sage cell
sagecell.makeSagecell({inputLocation: ‘#mycell’,
template: sagecell.templates.minimal,
evalButtonText: ‘Activate’});
// Make *any* div with class ‘compute’ a Sage cell
sagecell.makeSagecell({inputLocation: ‘div.compute’,
evalButtonText: ‘Evaluate’});


One of the neat things about Sage, the open-source computer algebra system is that you can easily embed it into any web page. Here is an example of some code that can be evaluated to plot a function and it’s derivative. For more information about Sage: http://sagemath.org. To learn how to embed Sage into your web page: http://aleph.sagemath.org/static/about.html.


Embedded Sage Cell

# You can put virtually any Sage code in this section, or simply edit the existing
# code any way you choose.
def f(x):
return x * sin(pi*x)

Applied Discrete Structures

Applied Discrete Structures by Al Doerr and Ken Levasseur, both from the UML Department of Mathematical Sciences, has reached two significant milestones:

  • The book is now listed as part of the American Math Institute’sOpen Textbook Initiative.
  • It has its first outside adoptions: classes at the University of the Puget Sound, Grinnell College, Casper College and Luzurne Community College are all using the book in the spring of 2013.
Resources, such as this dynamic demonstration of cyclic subgroups of the group of integers mod n are continuing to be developed. Move the sliders to see the different subgroups with varyingmoduli:

var cdf = new cdfplugin();
cdf.embed(‘http://faculty.uml.edu/klevasseur/ADS2/demos/cyclic_subgroups_mod_n.cdf’, 727, 873);

Creative Commons License
Applied Discrete Structures by Alan Doerr & Kenneth Levasseur is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License.