2008 - university of illinois · 2008 department of. this calendar was designed by tori corkery for...

College of Liberal Arts and SciencesUniversity of Illinois at Urbana-Champaign

2008Department of

This calendar was designed by Tori Corkery for the Department of Mathematics at the University of Illinois at Urbana-Champaign. A special thanks goes to Sara Nelson, Lori Dick, Debra Woods, and George Francis for their help and support during the creation of this calendar. The artwork shown in this calendar was created by our faculty and current and former students. My thanks to them for their contributions which made this calendar celebrating mathematical art possible.

Department of MathematicsUniversity of Illinois at Urbana-Champaign

1409 West Green Street, Urbana, Illinois 61801

[email protected] ■ www.math.uiuc.edu ■ Telephone: 217-333-3350 ■ Fax: 217-333-9576

Current Faculty Ahlgren, ScottAlexander, StephanieAndo, MatthewBalogh, JozsefBauer, RobertBergvelt, MaartenBerndt, Bruce C.Boca, FlorinBradlow, StevenBronski, JaredD'Angelo, John P.DeVille, R.E. Lee van den Dries, LouDunfield, NathanDutta, Sankar P.Duursma, IwanErdogan, BurakFord, KevinFossum, Robert M.Francis, George K.Füredi, ZoltánGhrist, RobertGorvett, Rick Haboush, William J.Henson, C. Ward

Herman, RichardHildebrand, A. J.Hinkkanen, AimoHundertmark, DirkIvanov, Sergei V.Jacobson, Sheldon Junge, MariusKapovich, IlyaKatz, SheldonKedem, RinatKerman, ElyKirr, Eduard-WilhelmKostochka, AlexandrLaugesen, Richard S.Leininger, ChristopherLerman, Eugene M.Li, XiaochunLoeb, Peter A.Malkin, AntonMcCarthy, RandyMerenkov, SergiyMiles, JosephMineyev, IgorMonrad, Ditlev

Muncaster, Robert G.Nevins, ThomasNikolaev, Igor G.Palmore, JulianRezk, CharlesReznick, BruceRosenblatt, Joseph M.Rosendal, ChristianRuan, Zhong-JinSchenck, HalSchupp, Paul E.Solecki, SlawomirSong, RenmingSowers, Richard B.Stolarsky, Kenneth B.Tolman, SusanTumanov, AlexanderTyson, JeremyTzirakis, NikolaosUhl, J. Jerry, Jr.West, Douglas B.Wu, Jang-MeiZaharescu, AlexandruZharnitsky, VadimZhu, Yanyun (Judy)

Department of

University of Illinois at Urbana-Champaign

www.math.uiuc.edu Jeremy Rouse is a J.L. Doob Research Assistant Professor who works in number theory. He is particularly interested in modular forms and elliptic curves. He joined the department in 2007. The image consists of 1000 small spheres. The spheres were placed by applying the spherical coordinates map to 1000 points on the line y = (8/9)x and were colored according to their placement in 3-space. Rouse created the source file using Maple and rendered the image using POV-Ray.

December

February

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1409 W. Green, Urbana, IL 61801 (217) 333-3350 email: [email protected]

New Moon First Quarter Full Moon Quarter Last

Sun Mon Tue Wed Thu Fri Sat

1 New Year’s Day

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20 21 Martin Luther

King Jr. Day

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Department of University of Illinois at Urbana-Champaign

www.math.uiuc.edu

Milos Curcic is a Ph.D. student in mathematics at the University of Illinois with an interest in combinatorics and graph theory, in particular. He is currently a NetMath teaching assistant and a research assistant for the College of Business working on a game theory project. In this graphic, the picture is superimposed on an arbitrary parametrized surface by a technique of texture mapping which uses surface parameters as the color coordinates in the image map. We then use linear algebra and matrix multiplication to calculate the coordinates of the surface relative to the position of the viewer and to transform a 3-D object into a 2-D picture on paper or screen. This particular example just involves parametrizing a surface and using this parametrization to match the points on the surface with the colors in the image map.

January

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March 4 5 1

6 7 8 9 10 11 12 3 4 5 6 7 8 2

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14 Valentine’s Day

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Presidents’ Day

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www.math.uiuc.edu

Scott Banister was a Chancellor’s Scholar in the Campus Honors Program at the University of Illinois at Urbana-Champaign from 1994–1996. Banister started his career as a pioneer in the email business. He was founder and Vice President of Technology at ListBot, the largest ASP for business email list hosting. Banister is now the co-founder and Vice President of Corporate Strategy at IronPort. This artistic detail is from Banister's 3-dimensional cellular automaton “Evolve”. This was part of his project for the CAVE immersive virtual environment taught by George Francis in Math 198 “Hypergraphics” 1995. Similar projects by prior students in the course implemented a 3-D version of Conway's “Life”. These convinced us that dense, asymmetrical patterns of identical 3-D shapes (markers) were difficult if not impossible to appreciate, even if you could float among them in full, stereoscopic immersion. Banister solved this perception problem by inventing a new cellular automaton, which was sparse (not like this print) and symmetrical. Later he was part of the team that created an ImmersaDesk Demo, “Cellular Semiotics” in computational biology for the 1995 Supercomputing Conference in San Diego. The lead mathematical visualizer on that team was math graduate student Alexei Bourd.

February

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New Moon First Quarter Full Moon Last Quarter


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Daylight Saving

Time Begins

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18 19 20 Vernal Equinox

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Easter

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www.math.uiuc.edu Valerie Peterson is a research/teaching assistant working with advisor Professor Robert Ghrist in the areas of applied topology and metric geometry. In particular, she uses tools from CAT(0) geometry to obtain results about robotic motion-planning problems. The figure shows three different views of a cube complex which, despite having well-behaved hyperplanes and the same link at every vertex, cannot be realized as a state complex for any reconfigurable system (a state complex is a cube complex which records all possible configurations of some moving system, as in robots traveling along a track). The image was created using XFig.

March

May 1 1 2 3

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Passover

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www.math.uiuc.edu

Ricardo Rojas is a graduate student in the Department of Mathematics at the University of Illinois at Urbana-Champaign. To complete his doctoral dissertation, he is using loop theory to study finite affine planes under the guidance of his thesis advisor, Professor Bruce Reznick. For a prime p, consider a square array of (p - 1)2 smaller squares. Fix an integer m which is not divisible by p, and color the square in the i-th row and j-th column black if ij is congruent to m mod p, and white otherwise. There will be exactly one black square in each row and exactly one black square in each column. With large primes p, interesting symmetries can result when m is varied; these can be enhanced when the colors are also varied.

April

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June 2 3 4 5 2 3 4 5 6 7 1

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Mother’s Day

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Memorial Day

27 28 29 30 31


www.math.uiuc.edu

May

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July

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New Moon First Quarter Full Moon Quarter

Faisal Mohamed graduated from Illinois with a Bachelor of Science in Computer Engineering in December 2003. He continued on to graduate school pursuing a Master of Science in Applied Mathematics under the advising of Karen Mortensen. Faisal obtained his Master's in December 2005. He currently works for Wolfram Research, Inc., developing and designing user interfaces for Mathematica. The graphics presented are based on a program he wrote while taking a Mathematica programming course at the U of I taught by Bruce Carpenter. It simulates the effect of looking at randomly generated polygons under a kaleidoscope. The simulation is achieved by cropping off a piece of the polygon and repeatedly reflecting it along the crop boundary to fill a circle. A point and a normal vector are used to represent lines used for cropping and as an axis of reflection. This line representation is essential as it contains direction information that can be used to decide which side of the polygon to keep after cropping.

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Last

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Father’s Day

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Summer Solstice

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www.math.uiuc.edu

Radoslav Kirov is a 4th year graduate student, working with Professor Iwan Duursma. Radoslav is originally from Sofia, Bulgaria and did his undergraduate work at Occidental College, Los Angeles, CA. He does computer graphics and web design as a hobby. In the summer of 2007, he wrote a software program that produces images such as the one shown here. Such "looped" effect was first done by Escher in his painting "Prentententoonstelling" although the mathematical nature behind it was probably unknown to him at the time. Amazingly, his painting has incredibly high precision in the depiction of the transformation, without the use of calculations. In the 2003 article "Artful Mathematics: The Heritage of M.C. Escher" published in the AMS Notices, B. de Smit and H. W. Lenstra Jr. explained how the transformation that produces the weird spiral-like recursion you are seeing can be described in mathematical terms. The "trick" behind the spiral is a multiplication map between elliptic curves over the complex numbers. To obtain the doubly-periodic elliptic curves from the original recursive image we use exponential and logarithmic maps.

June

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August 7 1 2

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Independence Day

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www.math.uiuc.edu

July

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Bruce Reznick and Zoltán Füredi are professors in the Department of Mathematics at the University of Illinois. Reznick has been on the faculty since 1979. He was a Sloan Foundation fellow from 1983–1986, and he received the Prokasy Award for Excellence in Undergraduate Teaching in 1997. Reznick’s research interests are in combinatorial methods in algebra, analysis, number theory, combinatorics and geometry, often involving polynomials. Füredi joined the department in 1991. His research interests are in the theory of finite sets with applications in geometry, designs, and computer science. If you plant 24 trees in this 4 x 6 irregularly-spaced rectangular array and sit at the center, you will notice that the trees appear to be exactly 360/24 = 15 degrees apart. This diagram was created in Mathematica : the x-coordinates are ± a, ± b and he ty-coordinates are ± c, ± d, ± 1, where a = √ 2 - 1, b = √ 2 + 1, c = √ 3 - √ 2 and d = √ 3 + √ 2. There is a similar 4 x 4 array, but no other 2m x 2n arrays with m, n ≥ 2 exist. This theorem was proved in the paper "The maximal angular gap among rectangular grid points" by Zoltán Füredi and Bruce Reznick, which appeared in the journal Periodica Mathematica Hungarica, volume 36 (1998), 119-137.

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www.math.uiuc.edu

This graphic was created using Mathematica by Eric Landquist, a graduate student who is studying arithmetic and class number computation in cubic function fields under Professor Iwan Duursma locally and Professor Renate Scheidler of the University of Calgary. This image represents an object called the infrastructure of the principal ideal class, and is a set of reduced principal ideals that almost forms a group under the operation of ideal composition (multiplication and reduction). The infrastructure exists in any number field or function field whose ring of integers has positive unit rank and is useful for computing regulators and fundamental units in these fields. In unit rank 1, by far the most understood case, the infrastructure is a cycle. In the unit rank 2 setting, however, the infrastructure is bicyclic like a torus, as the image shows. The spheres represent reduced principal ideals, the red and orange lines represent steps in one direction, and the blue lines represent steps in the other direction in the infrastructure. Specifically, this is the infrastructure corresponding to the cubic function field defined by the curve y 3 = x 6 + 5x 5 + 6x 4 + 5x 2 over 7.

August

October 1 2 1 2 3 4

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Labor Day

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Autumnal Equinox

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28 29 30 Rosh Hashanah


www.math.uiuc.edu

Jennifer Weber is a graduate student in mathematics at the University of Illinois. Her future interest in research lies in analytic number theory. This pattern is a representation of 1 of 17 different wallpaper patterns. There are only 7 unique strip patterns, and similarly, there are only 17 unique wallpaper patterns. The symmetries that each one has are a compilation from translation, to reflection, glide reflection, and rotation of either 60, 90, 120, or 180. This picture was done using Paint Shop Pro 9. As an honors project for non-Euclidean Geometry, she created her own sample of each of the 17 different wallpapers. Jennifer loves art and enjoyed using her art skills in math, but she says, “it was also challenging to create these to have only certain symmetries.”

September

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November 6 1

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Yom Kippur

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Columbus Day

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Halloween


www.math.uiuc.edu

Bruce Carpenter graduated from the University of Illinois in 1995 with a Ph.D. in mathematics under John Gray. He is currently a Teaching Associate in the Department of Mathematics at the University of Illinois at Urbana-Champaign. This graphic illustrates discrete steps in a linear homotopy, or morph. It starts with a square centered at the origin and deforms the sides of the square to produce an arrow (see inset). The amount of deformation on each side of the square changes with both angle and distance. The background highlights the tension between the continuous variation of color on the color wheel and the discrete stages of the morph.

October December 1 2 3 4

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Daylight Saving

Time Ends

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Veterans Day

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Thanksgiving

28 29

Department of

University of Illinois at Urbana-Champaign

www.math.uiuc.edu Alexei Bourd received his Ph.D. in mathematics in 2003 from the University of Illinois under the direction of Professor Jared Bronski. He was a research assistant for Professor George Francis in the Renaissance Experimental Lab of the NCSA, and our Apple Lab, and a TA for Math 198 "Hypergraphics" in 1995 and 1997. He joined Qualcomm in 1997, where he is now senior staff engineer. This detail of a view inside a model of a bilipid layer membrane was recreated from Bourd's C/IrisGL software, celsem.c, specially for this calendar. Celsem was the visualization engine for “Cellular Semiotics”, an immersive virtual environment (CAVE) extravaganza that was shown at Supercomputing ‘95 in San Diego. Cellular Semiotics was the collaboration of two teams of molecular biologists, at Houston and the NCSA, and Francis's CAVE visualization team that included Bourd's student, Scott Banister. In close collaboration with Marcus Wagner, who parallelized the Houston work on NCSA supercomputers (which supplied real-time data to San Diego), Bourd designed and built celsem. Bourd supplied the underlying wave-like motion for the membrane (PDEs), and the docking (differential geometry) for the allergen-antigen pairs of molecules adhering to the membrane. The picture you see shows an ingenious combination of real-time molecular models inside virtual cells with trompe-l'oeil wall-paper (texture mappings) of the scene in neighboring cells. This optimization of effort made real-time CAVE navigation a reality.

November

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Winter Solstice

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Hanukkah

23 24 25

Christmas

26 27

28 29 30 31

2008 - university of illinois · 2008 department of. this calendar was designed by tori corkery for...

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