Math is all around us! Don’t miss Fields Medalist Manjul Bhargava’s eight-week minicourse on math and magic — a perennial favorite of the Princeton crowd — or the season opener of *Quadrivium*, MoMath’s Math plus Music Salon hosted by Marcus Miller and featuring Latin percussionist Bobby Sanabria (seven-time Grammy Award nominee) and drummer Andrew Atkinson (*Hamilton* on Broadway). For the literary crowd, there’s a triple play of author events, with astrophysicist Brian Keating, bestselling author Jim Holt, and UK mathematician Hannah Fry all coming to discuss their books. And back to school means back to math for students, too, who can explore the intersection of math, design, and computer programming in the all-new *Symmetry Code* workshop, sit back and enjoy the comedy of *Math on Stage* with Matt Parker and friends, or explore all-new games and activities during MoMath’s teen-only social night. All this and more are in store this fall at the museum America’s buzzing about: buzzfeed.momath.org.

Tue, Oct 2 |
Maths Inspiration, Live on Stage! featuring math comedian Matt Parker |

Tue, Oct 2 |
Live at MoMath: Matt Parker returns! — an evening of math and stand-up comedy |

Wed, Oct 3 |
Math Encounters: “Calculated Risk: Living is a Dangerous Business” with Jen Rogers |

Sat, Oct 6 |
Global Math Week kickoff celebration in San Jose, CA |

Wed, Oct 10 thru Tue, Oct 16 |
Global Math Week — help us reach 10 million people! |

Thu, Oct 11 |
The Math of Particle Physics, with CERN researcher James Beacham |

Fri, Oct 12 |
Family Fridays at MoMath presented by Two Sigma: “No Duck Can Waltz! — Lewis Carroll’s Logic Puzzles” with Chaim Goodman-Strauss |

Sun, Oct 14 |
Tween Primes, the MoMath book club for tweens and teens: Letters to a Young Mathematician by Ian Stewart |

Sun, Oct 14 |
Unlimited, MoMath’s mix-n-mingle program for students in 6th through 9th grades |

Thu, Oct 18 |
Volumes, the MoMath book club |

Sun, Oct 21 |
“Catch that Ghost” — a MoMath puzzle hunt throughout midtown Manhattan, especially for families |

Mon, Oct 22 |
Po-Shen Loh, coach of the three-time gold medal USA IMO team, returns to MoMath for a special presentation |

Tue, Oct 23 |
2018 MoMath Gala, hosted by Peter Muller and featuring retired NFL professional John Urschel |

Wed, Oct 24 |
Math Gym, a workout for your brain! |

Mon, Oct 29to Wed, Oct 31 |
MATRIX 2018: MoMath’s biennial math outreach conference in Barcelona, Spain |

**Follow MoMath on Social Media**

**New this fall! Math Gym, a workout for your brain!**

Students, spend an hour at MoMath independently working on engaging and beautiful math problems, hand-selected by MoMath’s advisory council of math PhDs. Choose whichever problem you like from a specially curated selection and work under the guidance and with the mentorship of one of MoMath’s expert PhD mathematicians. If you love math and want to experience the incredible joy of mathematical discovery, you won’t want to miss this new monthly program. Dates will vary from month to month; program debuts September 26 and returns the following month on October 24. Register today at workout.momath.org.

*Symmetry Code*: Exploring Symmetry Using the Wolfram Language

Evening options (Friday, September 14 OR Thursday, September 27, 6:15 pm)

Weekend options (Sunday, September 16 OR Saturday, September 22, 9:00 am)

Students in grades 7 through 12 are invited to a three-hour course during which they will learn how computer programming can be used for explorations into mathematical topics. Using the Mathematica programming language, students will explore the topic of symmetry and introductory computing, plus take home a t-shirt with their very own MoMath logo design! Learn more and register at symmetry.momath.org. Scholarships available for students in need, courtesy of our friends at BNY Mellon.

*PolyPals*

Weekly, beginning in September

*PolyPals* is back! The popular MoMath program for toddlers and preschoolers returns. Recent studies have shown that a child’s math skills upon entering kindergarten can be a strong predictor of future academic performance in both math and reading throughout the elementary grades. Math learning promotes working memory, improves attention, and increases other basic cognitive skills. *PolyPals* offers the opportunity to listen to math-themed stories and have fun exploring the concepts behind them. Engage your toddlers and preschoolers in these playful activities to help them develop a strong foundation. Children will participate in songs, stories, and activities tied to a different theme each week. Patterns, shapes, numbers, and more — *PolyPals*is a great way to enrich your child’s day with mathematical inspiration. Details and registration coming soon to polypals.momath.org.

**Apply now to Expansions: MoMath’s innovative program to engage and challenge NYC’s brightest math students **

Reinvent math class with

**LAST CHANCE TO APPLY! High school students: Volunteer at MoMath on weekends during the 2018-2019 school year**

High school students with a love of math can become a part of MoMath! Gain valuable experience while helping others and add something unique to your college applications. Apply by midnight Monday, September 4 to be considered for a fall semester start: volunteers.momath.org.

**Weekend programs for families**

Take a tour with MoMath’s *Derivatives* program (derivatives.momath.org) or join one of MoMath’s specially-trained educators in *Explorations*, a hands-on classroom experience to discover the wonder of mathematics (explorations.momath.org). Don’t miss your chance to see math in a whole new light, only at MoMath.

**Retail Pick of the Month — Dodecahedron sticky memo ball**

Keep track of your memos in a mathematically stylish way: the Dodecahedron sticky memo ball is a functional desktop memo center with a pad of sticky notes on each of its 12 pentagonal sides. Use up all the notes to reveal a fun mathematical message.

**Maths Inspiration, Live on Stage! Featuring math comedian Matt Parker
Tuesday, October 2, 10:00 am at the Baruch Performing Arts Center**

Educators, do your students wonder, “What’s the point of algebra? Who on earth needs Pythagoras? And who cares about exponents, anyway?” Inspire your students — even the ones who don’t love math — with a large venue stage show on October 2 featuring some of the UK’s most engaging math performers. Learn more and register at stage.momath.org. Special thanks to OppenheimerFunds and Bank of America for making this event possible.

*Live at MoMath: Matt Parker returns!* — an evening of math and stand-up comedy

Tuesday, October 2, 7:00 pm

Join MoMath for a night of laughs as YouTube phenom Matt Parker, stand-up comedian and mathematician, returns with his unique brand of math-based humor. Don’t miss this night of mathematical comedy; join Matt as he approaches an infinite amount of humor (but never quite gets there). Learn more and register at standup.momath.org.

*Math Encounters*: “Calculated Risk: Living is a Dangerous Business” with Jen Rogers

Wednesday, October 3, 4:00 pm and 7:00 pm

Real life involves risks, from cycling without a helmet to visiting a country with contagious diseases to living near the water. Sometimes your perception of risk is distorted, making you think that some things are more — or less — risky than is actually the case. How statistically significant are the risks that we see? And what do the numbers mean? Jen Rogers, Director of Statistical Consultancy Services for the University of Oxford, demonstrates how statistics can be used to make better decisions about risky activities. Learn more and register at mathencounters.org.

**Global Math Week kickoff celebration
Saturday, October 6 at Santa Clara University**

Join MoMath and the Global Math Project for the kickoff of Global Math Week 2018. The day starts with a symposium at Santa Clara University including events and talks for families, students, and teachers. The fun continues that evening with a celebration at The Tech Museum in San Jose. For more information, visit globalmathproject.org.

**Global Math Week
Wednesday, October 10 through Tuesday, October 16**

MoMath is proud to be one of the organizing partners of Global Math Week. Help reach millions of students with

*The Math of Particle Physics*, with CERN researcher James Beacham

Thursday, October 11, 6:30 pm

Don’t miss James Beacham’s visit to MoMath this October 11. As a scientist at CERN’s Large Hadron Collider (LHC) and a member of the ATLAS collaboration (one of the teams that discovered the Higgs boson in 2012), James searches for evidence of new particles — dark matter, gravitons, dark photons, and exotic Higgs bosons among them — that could provide answers to the fundamental open questions of physics. Register now at particle.momath.org.

*Family Fridays* at MoMath presented by Two Sigma: “No Duck Can Waltz! — Lewis Carroll’s Logic Puzzles” with Chaim Goodman-Strauss

Friday, October 12, 6:30 pm

Did you know that every flying pig is purple? This is absolutely true, at least logically, for the simple reason the statement can’t be false — there is not any flying pig that isn’t purple! Come explore the ins and outs of mathematical logic, with puzzles from Lewis Carroll’s books and other absurd examples. Learn more and register at familyfridays.momath.org.

*Tween Primes*, the MoMath book club for tweens and teens: *Letters to a Young Mathematician* by Ian Stewart

Sunday, October 14, 5:30 pm

*Letters to a Young Mathematician* tells readers what Ian Stewart wishes he had known when he was a student and young faculty member. Subjects include what mathematics is and why it’s worth doing, the relationship between logic and proof, the role of beauty in mathematical thinking, the future of mathematics, how to deal with the peculiarities of the mathematical community, and many others. This book is sure to be a favorite with students at all levels! Learn more and register at tweenprimes.momath.org.

*Unlimited*, MoMath’s mix-n-mingle program for students in 6th through 9th grades

Sunday, October 14, 6:30 pm

Don’t miss the next installment of *Unlimited*, MoMath’s mix-n-mingle evening for students in late middle school through high school. If you’re excited by the idea of a fun-filled, math-focused night where you can hang out with old and new friends, enjoy cookies and hot chocolate, and move to some great tunes, then you won’t want to miss this exciting program on October 14. Registration coming soon to unlimited.momath.org.

** Volumes, the MoMath book club
Thursday, October 18, 6:30 pm
**MoMath favorite Hannah Fry has a new book! In HELLO WORLD, Hannah explores the role of algorithms in automating important decisions in crime, healthcare, relationships, and money. Should we entrust ourselves to driverless cars? Are machines capable of producing great works of art? How do we balance the importance of making better medical diagnoses with the value of preserving privacy? Our increasing reliance on pre-defined, quantitative decision-making rules, some of which are invisible to us, raises some interesting questions. If you enjoyed Hannah’s talk in September and want the opportunity to delve more deeply into some of the issues she exposed, or if you just want to join a provocative discussion of an increasingly serious issues facing society today, you won’t want to miss this event. Visit volumes@momath.org.

*Equilibrium*, an adult evening of mathematical games

Saturday, October 20, 6:00 pm

Tabletop gaming is more fun than ever! Come join old friends and new for a fun-filled, adult evening of mathematically rich, hands-on games. Bring a snack, play some games, and connect with new and interesting people, all while enjoying the unique evening atmosphere at the nation’s only Museum of Math. Learn more and register at equilibrium.momath.org.

**“Catch that Ghost” — a MoMath puzzle hunt throughout midtown Manhattan, especially for families
Sunday, October 21, 11:00 am **

In this family-friendly puzzle event, teams of children and adult chaperones will explore midtown Manhattan on a location-based adventure, using their wits and mathematical knowledge to solve a mystery! This puzzle hunt is specially designed for families and solvable in approximately three hours. Learn more and register at puzzlehunt.momath.org.

**Save the date! Po-Shen Loh returns to MoMath for a special presentation
Monday, October 22, 6:30 pm**

Po-Shen Loh, professor of mathematics at Carnegie Mellon University and coach of the three-time gold medal-winning USA International Mathematical Olympiad team, returns to MoMath for one night only. Watch events.momath.org for details, coming soon!

**2018 MoMath Gala, hosted by Peter Muller and featuring retired NFL professional John Urschel
Tuesday, October 23, 6:00 pm**

Calling all mathletes, athletes, and dedicated fans! Join special guests including mathematician and retired NFL professional John Urschel, IMO Team USA coach Po-Shen Loh, Fields Medalist Manjul Bhargava, and many other acclaimed mathematicians at MoMath’s annual gala, a fundraiser to support the Museum. Learn more and register at gala.momath.org.

*Math Gym*, a workout for your brain!

Wednesday, October 24, 3:30 pm

Students, spend an hour at MoMath independently working on engaging and beautiful math problems, hand-selected by MoMath’s advisory council of math PhDs. Choose whichever problem you like from a specially curated selection and work under the guidance and with the mentorship of one of MoMath’s expert PhD mathematicians. If you love math and want to experience the incredible joy of mathematical discovery, you won’t want to miss this new monthly program. Register today at workout.momath.org.

**MATRIX 2018
October 29-31 in Barcelona, Spain**

Are you a math educator, mathematician, or science museum professional interested in math outreach? If so, the 2018 MATRIX conference may be for you! Join MoMath and our colleagues from MMACA and other math museums around the world for the third international conference on math outreach, taking place in lovely Barcelona, Spain, this October 29 through 31. For more information or to register, visit matrix.momath.org.

**MoMath 2018-2019 Professional Development Days**

Educators, register for one, two, or three professional development days at MoMath during the 2018-2019 school year. Advance tickets are $175; day-of tickets will be $199, if available. **Special offer:** Buy all three days together for a discounted price of $495!

*Metros and Metrics* — Professional Development Day

Tuesday, November 6, 9:30 am

Educators, mark your calendars now for an exciting day of mathematical discovery this fall. Taxicabs, the NYC subway, and the British rail all serve as a backdrop for an exploration of bizarre alternative concepts of distance. Don’t miss this unusual day of hands-on professional development on November 6. Learn more and register at pd.momath.org.

*M-cubed* — Professional Development Day (high school educators only)

Monday, January 28, 2019, 9:30 am to 3:00 pm

Join Professor Alex Kontorovich as he illuminates some of the most beautiful and timeless mathematical problems and solutions, the people behind these discoveries, and the historical developments that led to each breakthrough. Geared for high school educators, this series will provide a bird’s-eye view of many landmark developments in mathematics, from antiquity to the 20th century. Learn more and register at pd.momath.org.

*Artful Math* — Professional Development Day

Thursday, June 6, 9:30 am – 3:00 pm

Connect the dots between math and art! Spend the day gaining new perspectives as you enjoy the creative side of math, then turn things around and explore the analytical side of art. Plus, experience engaging, hands-on activities at the intersection of math and art that you can take back to school to share with your students. Learn more and register at pd.momath.org.

*Math Encounters*: “Certain Variation: You Never Step in the Same River Twice” with Gerald van Belle

Wednesday, November 7, 4:00 pm and 7:00 pm

Change is the one constant in life. How can mathematics bring some certainty to variation? Biostatistician Gerald van Belle shares everyday stories and interactive activities that highlight the sometimes surprising analysis of variability. Reinforcing planes in warfare, walking or running in the rain, storks and babies, missing data, and more — see how a mathematical lens helps shed new light on the study of change. Special introduction by Gerard T. van Belle, Chief Scientist of Lowell Observatory’s Navy Precision Optical Interferometer. Learn more at mathencounters.org.

**Save the date! Special guest appearance by Grant Sanderson
Tuesday, November 13**

MoMath is pleased to announce a special appearance by Grant Sanderson of 3blue1brown, one of the hottest YouTube math channels around, with 42 million views and counting. Watch events.momath.org for details, coming soon!

*Family Fridays* at MoMath presented by Two Sigma: “Recycled Math” with Nick Sayers

Friday, November 16, 6:30 pm

Ever seen art made from trash bags, realtor signs, plastic bottles, or soda cans? Join science-inspired British artist Nick Sayers, who makes art from all these everyday materials and more. See examples of his work and enjoy some hands-on activities demonstrating how mathematics underpins it all. Don’t miss this opportunity to explore the hidden geometry of trash, the magic of mathematics, and the creative potential of recycling at familyfridays.momath.org.

**Events, birthday parties, and more**

Looking to host a one-of-a-kind event where your guests can interact with over 40 engaging exhibits? Enter a world of mathematical intrigue, but don’t worry: amidst all the activity, there is plenty of space for gala-worthy dinners, over-the-top birthday bashes, laser-cutting parties, and bar/bat mitzvahs. Who knew math could be this much fun? Email events@momath.org for more information.

**School and group visits**

MoMath has over a dozen great programs for school groups visiting the Museum. From graph coloring to Möbius bands, bring your students to MoMath for a view into the exciting world of mathematics and see why kids of all ages love visiting the Museum. Register at fieldtrips.momath.org.

**Free trips for Title I schools**

Thanks to the support of companies including OppenheimerFunds, Con Edison, and Two Sigma, as well as some generous MoMath friends, support for Title I schools is now available. To apply for a free trip in the 2018-2019 school year, visit titleone.momath.org. Interested in sponsoring a field trip? Email donation@momath.org.

**Beautiful Math**

See what mathematicians think is beautiful about mathematics at beautiful.momath.org. Now featuring Stephen Wolfram!

**Have you tried out for Varsity Math yet?**

Get your weekly puzzle fix from MoMath in the

**Spread the MoMath word**

Like MoMath? Let the world know! Share your comments on:

- Yelp: yelp.com/biz/museum-of-mathematics-manhattan
- Facebook: facebook.com/MoMath1
- TripAdvisor: tripadvisor.com/Attraction_Review-g60763-d3726207-Reviews-National_Museum_of_Mathematics-New_York_City_New_York.html
- Foursquare: foursquare.com/v/museum-of-mathematics-momath/4e4e8fa081308c328c67daae

**Join the MoMath community**

Become a member today and help ensure that MoMath continues to deliver exciting and engaging math programs for all ages. Visit momath.org/join to become a member and receive unlimited access to MoMath’s innovative exhibits, plus discounts in *Additions*, the shop at MoMath. Become a premium member and receive early notices and invitations to exclusive MoMath events. Join now and take advantage of MoMath’s low rates. To learn more about long-term memberships, call (212) 542-0566.

**We need you!**

Interested in volunteering on the Museum floor? If you love math, would like to help others enjoy MoMath’s interactive suite of exhibits, and are willing to devote two 4-hour shifts each month, please send an email to jobs@momath.org with the subject line “MoMath integrator.” Please include a cover letter, current résumé, and a newly-written essay that, in approximately 150 to 300 words, describes an experience that shaped your love of mathematics.

**We hope to see you at MoMath!**

As you walk toward this striking mosaic by artist Miguel Covarrubias, invite your tour participants to look at the boundaries between regions of color in the mosaic, and identify one high-contrast boundary and one lower-contrast boundary. Then, invite the participants to come up very close to the mosaic and examine how the tiles look in the vicinity of each of the boundaries that they identified. They should see that at the high-contrast boundary the edges of the individual tiny tile lie right along the boundary, whereas at the low-contrast boundary, the tiles don’t seem to have any special relationship with the boundary and that up close, the boundary may seem to “dissolve” and be very hard to pin down a precise location for, even though the boundary was clear from far away. These observations show how the macroscopic characteristics of the image affect the microscopic characteristics of the tiny tiles that make up the image. As we’ve discussed with

Before you step away from the close-up view of the mosaic, have the participants look at one region of fairly uniform color and examine it closely. Then have them step far back from the mosaic, say under the awning of the entrance to the museum, and look again at the same region of the artwork. From here, the region should look like a uniform patch of color, whereas up close you could see that there were individual tiles of different colors. The colors have blended to create just the shade the artist desired. Sometimes we call this “adding” one color to another. Is this a kind of adding that we can understand mathematically, or is it just a sort of analogy? Actually, it is something that we can make mathematical models of. An old and fairly simple model is an artist’s color wheel, such as this one.

You can interpret this wheel as representing colors with angles around a circle; and if you want to find the result of adding two colors, you take their angle bisector. For example, yellow is at 90 degrees and blue is at 210 degrees and the green you get by mixing them is halfway in between, at 150 degrees. There are many more combinations that work this way as well, such as yellow at 90 degrees mixing with orange at 30 degrees to produce marigold at 60 degrees.

But as with any mathematical model of the real world, this angle model of color has its limitations; it can be pushed to a point where it begins to break down. For example, consider combining yellow at 90 degrees with purple at 270 degrees. That’s a straight angle in this diagram, so immediately you have a question of which of two equally plausible angle bisectors you ought to use, the left or the right? And further, neither really seems to give a satisfactory answer; it seems dubious that yellow and purple would combine to make either red-orange or cyan.

But also as with any model, you can investigate more deeply and refine it further, possibly at various levels of depth and detail. So for example, if you are building a color printer, and need to determine just how much of each color of toner to use to reproduce the color that the computer user wants to see on the printout, then you’re going need a more precise model of color combination, such as the Kubelka-Munk equation, shown below. While we’re not going to go into the details of this equation here, it serves to illustrate how math is a tool that can be used to model and understand the world around us in a huge variety of ways and at a great range of levels, touching on nearly every aspect of life, including the aesthetic and artistic worlds.

As you walk from Klyde Warren Park to the Dallas Museum of Art, you will pass by the Hunt Oil building. Its facade features the intersection of two cylinders, a narrower vertical one and a much larger-radius horizontal one. The curve where the two surfaces intersect is highlighted in silver trim. This is a beautiful, sweeping curve that does not lie in any one plane in space. It does not correspond to any familiar, simple curve; it’s not an ellipse or parabola or helix, etc. Such curves have been studied by mathematicians, however, and are called Steinmetz curves. It’s wonderful how math shows us that beautiful complexity can arise even from simple combinations of very simple shapes, such as these cylinders. ]]>

In a park filled with a number of mathematically interesting sights (such as the row of arches along one edge), this climbing structure is arguably the most mathematical object of all. You’ll notice that it is constructed of metal rods joined at the spherical red hubs (with black circles on them). If you look at it further, you will see that exactly five rods meet at every hub, and that every three adjacent hubs are connected in a triangle. So what we have is a structure composed of identical faces, each of which is an regular polygon (in this case, an equilateral triangle), with the same number of faces meeting at every vertex. That’s a very special set of conditions that guarantees that the structure is very symmetric and regular, and it turns out that only five such structures exist, the so-called

If you’re looking for more material, you can count the number of faces (20), bars (what a mathematician calls *edges*) (30), and hubs (which a mathematician would call *vertices*) (12). Notice that faces plus vertices differs from edges by only two. If you have models of other polyhedra, participants can compare faces plus vertices with edges, and they will always find the difference is two. This remarkable fact is called Euler’s formula, and it applies to any polyhedron that is roughly speaking ball-shaped (as opposed to doughnut-shaped).

Above I claimed that this is perhaps the most mathematical object of all in the park, and so I want to show one specific sense in which that’s true. What is the most amount of symmetry that any three-dimensional object can have? Well, it’s the symmetry of the sphere, in which any rotation whatsoever leaves a sphere unchanged. (Good to illustrate this with a ball or other spherical object.) Do you think there’s a second-most amount of symmetry any object can have? It turns out there is. If you look in the center of the climbing structure, amidst the network of ropes to climb on, the centermost cell of the structure is a polyhedron made up entirely of pentagons and hexagons. You might call it a soccer-ball shape, because it is the same as the traditional pattern of panels on a soccer ball; or maybe a chemist would call it a “buckyball” because it is the same shape as a molecule consisting of 60 carbons that goes by that name; but a mathematician would call it a *truncated icosahedron*. Let’s look into how much symmetry it has. (This is best done with a model.) You can hold it up so that one vertex is at the top and the pentagon that touches that vertex is facing forward. Then have one of the participants choose any other vertex, and show how you can move that vertex to be at the top with the pentagon facing forward. So there is one symmetry for each vertex, and there are 60 vertices. In addition, this polyhedron is mirror-symmetric (looks the same when reflected in a mirror). That doubles the number of symmetries, because any of the rotations can be combined with a reflection.

Thus, this object has 120 different symmetries, which turns out to be the greatest number of symmetries that any object (other than a sphere) can have. Kids get to play in, around, and on it, absorbing its natural mathematical form. (I should mention that the outer structure, the icosahedron, also has 120 symmetries; it’s just much harder to see what they all are and count them with the icosahedron than with the truncated icosahedron,)

Speaking of Platonic solids in public places, there’s at least one more of them easily visible from this spot. It’s the Perot Science Center, which is in the shape of a very large cube: identical square faces, with three meeting an each corner (or vertex). You might also ask what the most significant departure is from it being a perfect cube. I think that would have to be the large black structure slanting up the side. Why is it there? It houses escalators to get people to the higher levels of the building. This is another example in which concerns about human accessibility affects the mathematics of the structure.

]]>What you can observe, especially in the upper right of this photo, is a series of similar shapes formed by the veins of the leaf, repeating on smaller and smaller scales toward the edge of a leaf. This type of pattern with the self-similarity of the same motif repeating at a series of scales is called a

The basic mechanism at play here is that the tree needs to provide circulation to all parts of the leaf, to provide nutrients and to collect the products of the leaf’s photosynthesis. That circulation travels along the veins of the leaf. But a vein is basically a linear, or 1-dimensional structure; whereas the leaf spans a surface, or a 2-dimensional structure. So there seems to be a fundamental mismatch, and the tree would seem to have a problem nourishing the entire leaf.

However, as the image below shows, it is possible to twist, wiggle, or fold up a linear structure (in this case, a strip of paper) so that it starts to cover or at least come near nearly every point across a surface. This particular structure is known to mathematicians as the “dragon curve”, and if we made finer and finer versions of it with more and more folds it would come closer and closer to actually covering the surface of a table.

When you take a linear structure and fold it like this, it starts to take on some characteristics of the higher dimensional structure that it is occupying. So much so that, in a certain mathematically measurable and precise way, it takes on a fractional dimension between 1-dimensional and 2-dimensional. That’s where the word “fractal” comes from — it refers to the FRACTional dimensionALity of these types of structure.

That’s why you see them in natural settings like this leaf. The tree, faced with the task of circulation to a 2-dimensional leaf using 1-dimensional veins, has naturally evolved a repetitive, self-similar structure spanning multiple scales, i.e. a fractal, to bridge the gap between these dimensions.

]]>Among other things, math is a tool we can use to understand the natural world. To do that, we need to start with data — often measurements of things we find around us. You and the folks on your tour can try your hand at it right here with the bamboo growing at the corner of the Nasher sculpture center. What are some aspects of the bamboo that we could measure? There are lots: number of leaves on a stalk, height of the first leaf on a stalk, circumference of a stalk, separation of the “joints” on a stalk, and many more. So choose a couple with your participants, and get them to generate data. (I would limit it to just two variables of your choice.) Then you can eyeball the data, or even make a quick, rough scatterplot of the data on your portable whiteboard, to see if there appears to be a relationship between the variables. For example, it appears that the stalks that have longer segments also have larger circumferences. Then you could discuss what the relationship you find, if any, tells you about the plant.

Turning from the natural to the built world, you can tell the tour participants that in the construction of this building, the architect Renzo Piano felt there was one geometric ratio that produced the most pleasing, harmonious forms, so he used it over and over again throughout numerous different features of the building and its rooms. You can ask them to guess what ratio that might have been. You may get guesses of the golden ratio, simply because that is often touted as a ratio that recurs in architecture, or you may get other guesses, or no guesses at all. In any case, once you’ve tried the guessing route, you should suggest that the folks on the tour discover the ratio by measuring things. You will all soon find that the ratio is 2:1, and you can find it in the stones that make up the walls, the paving stones on the sidewalk outside the building, the bays in the front of the building (ignoring the slight arc at the top, they are 32 feet wide and 16 feet tall), and in many other places. I should note that if you look at the main doorway inside one of the bays, you will find that they are 10 feet tall in a half-bay that is 16 feet wide. That’s *not* a 2:1 ratio, and in fact it is pretty close (between 1% and 2% off) to the golden ratio. That could be a coincidence, or it could be a sly nod to those who adulate the golden ratio in architecture.

You may want to pause to admire the Meyerson Symphony Hall across the street from it, where you can appreciate the architecture by I.M. Pei which deliberately uses the motifs of square and circle in opposition to each other to create dramatic tension. Look for all of the places where you can see circles inside of squares and vice versa. You may also want to take note of the window framing that slants up the side of the curved windows on the left-hand side of the building in this view. Since that window has a larger circular footprint at its bottom than an its top, it is a section of a cone, and the framing appears to have a constant rate of climb up the cylinder as it wraps around. That would make that curve in space something known as a a

If you look down at the ground near the side of the street, opposite the Meyerson, you will see that it is paved with thousands of small square stones. On the other hand, the surface has numerous undulations and contours, sculpted by natural forces such as the growth of tree roots, the running of water in the earth beneath the stones, and presumably a host of other natural forces. Yet nothing over the years has changed the fact that the top surface of each stone is perfectly flat and perfectly square. So building on what we saw on the Texas Sculpture Walk in the Jesus Moroles sculpture, a complex curved surface in space can be realized (or at least approximated) as an assemblage of innumerable small flat elements. This perspective on curved surfaces is a very powerful tool to analyze and understand them, and lies at the heart of what’s known as multivariable calculus. But we can appreciate it here as the duality between the natural landscape that exists in reality and the regular grid of flat squares that human engineering has inscribed thereupon.

Especially on a tour with younger kids, I feel like it’s often a good idea to highlight one really big number. The grid of stones here provides an excellent opportunity for that. You can highlight that mathematics isn’t only about getting the exact answer, it’s also about estimating, and that all we want to do here is estimate the number of stone blocks. So we can ignore a few blocks missing here or there or slight irregularities in the pattern. You can discuss strategies for estimating the total number of blocks. Most likely you’ll want to treat the grid as a big long, thin rectangle of blocks, so you need to know the number of blocks in each direction, and then multiply. In one direction it may make sense to count directly (perpendicular to the street), but not in the other direction, along the street, because the grid runs the length of an entire block. Instead, you might use multiplication again, noting that the sidewalk has giant blocks that are pretty regular. So you can have participants count the number of the giant sidewalk blocks and also how many little stones there are per block. Since it’s just estimation anyway, I am not going to include any numbers here, other than that you should end up somewhere between ten and twenty thousand individual blocks. To highlight the size of that number, you can ask the crowd to guess how tall a pile they would make if they were all stacked in a single column, one on top of the other. You can guess they are each about two inches thick (I didn’t see any with their sides exposed, but if you do, you can measure.) So take the estimate of the number of blocks, double it to get inches, divide by 12 to get feet, and then divide by 5280 to get miles; I came up with a stack over a half mile high.

]]>The Texas Sculpture Walk is such fertile ground for bringing out mathematical themes in contemporary sculpture that we could do an entire tour just within this one section of the District. We’re not going to indulge in that right now, to keep this set of tour ideas broadly based in the whole arts district, but I can’t resist mentioning just one of the sculptures, pictured above: *Spirit Inner Columns* by Jesus Moroles, just at the top of the stairs leading up to the Walk. In the picture, the sculpture might appear to be just three stacks of rectangular pieces of stone. However, in person, you can see that between each rectangular slab there is a smaller piece of stone, and taken together those smaller pieces create a graceful, flowing curve hidden amongst the uniform slabs. Each of the smaller pieces itself is really just a slab of stone, so this artwork shows how a curve can be understood as an assemblage of very numerous infinitesimal pieces, each on its own essentially straight and featureless. This is precisely the same insight that allowed Archimedes to compare the volumes of a sphere and cylinder over two millennia ago, and which underlay the invention of calculus in the 17th century. It underscores how both art and math can be about seeing the world with fresh eyes that will bring out heretofore unrecognized structure and detail.

It’s worth stopping at the landing halfway up to the Texas Sculpture Walk to show participants this image.

Tell them that this is a mathematical fossil, and that you’ll explain what you could possibly mean by that when they have found this same pattern somewhere nearby. It won’t take them long to find it in the bottom surface of the fountain/artificial stream that runs by this landing. So what is this image? It is the fossilized remnant of a prehistoric streambed, and it shows the same ripple patterns that you can find in almost any stream today that has a sandy bottom. There’s a very characteristic shape to the ripples: very shallow slope on the upstream side, and a much steeper slope on the downstream side. These patterns arise because of the dynamics of waves in moving water. A very significant topic in mathematics is the analysis of wave phenomena, and that type of mathematics can be used to understand and derive the ripple patterns seen in this fossil, proving that the math we do today was equally valid millions of years ago. Notice that not only do the waves in a stream mold the streambed into this shape, but this shape affects how the water flows by, so the engineers that created this fountain can use the results of this mathematics to create a more naturalistic environment tucked within the urban setting of the Dallas Arts District.

There are two main centers of mathematical investigation of the Wyly Theatre highlighted here.

First, if you walk right up to the corner of the building (the near left corner in the picture above) and (on a bright, sunny day) look straight up at the corner amidst the aluminum pipes which adorn the building, and maybe move your head back and forth a bit, you should be able to catch glimpses of bright sun or sky *between* the pipes and the building. Why would that possibly be? Why wouldn’t the engineers just fasten the pipes tight to the front of the building? The answer lies in the phenomenon of *thermal expansion*. For any object, there is a linear relationship between the change in length of the object and the change in temperature. You can find that relationship using what’s called the coefficient of thermal expansion, which is based primarily on the material the object is composed of. You multiply the original length of the object, times the coefficient of thermal expansion, times the change in temperature to get the change in length. For aluminum, the coefficient is approximately 0.0000123 (the units of that coefficient are inverse degrees Fahrenheit, for the record). So let’s try to see how much the pipes might change length between the coldest and hottest day in Dallas.

First, how long are the pipes? They run all the way from the roof of the building, which we can look up to find out is 132 feet tall, to the top of the first story, which we can eyeball to be somewhere in the vicinity of 16-20 feet high. So let’s call the pipes 115 feet long, for simplicity; remember, we are just making an estimate. Since the coefficient of expansion is so small, it will be easier to multiply by 12 to get their length in inches. So let’s use an estimate that the pipes are 1,380 inches long.

Next, what could the temperature swing be from the coldest to the hottest day in Dallas? At the time I first gave this tour, there had recently been a low of 18 degrees Fahrenheit, about the coldest anyone could remember for Dallas. In the summer, though, it can get over 100 degrees easily. So let’s say there could be a 90-degree swing between the coldest and hottest day. Then the pipes might change length by 1380×0.0000123×90 inches, which is approximately 1.5 inches.

That might not seem like much, but imagine if the pipes were bolted to the building at the top and bottom. If the separation between those bolts needed to change by 1.5 inches, it would shear the bolts right off, and there might be pipes falling down around the entrance of the building — not exactly a safe situation. So the architects had to allow the pipes to slide up and down a bit in grooves that keep them close to the building, but do not anchor both ends of the pipe. Those grooves create the space through which you can catch a glimpse of sky. Here we see math being used to keep our public buildings safe.

Now, it’s worth pointing out that the building itself will be contracting and expanding with the change in temperature. So why doesn’t that compensate for the expansion of the aluminum? The answer is that the pipes, being nearly pure aluminum, have a very different coefficient of thermal expansion than other materials used in the building. For example, the coefficient of thermal expansion of glass is only about 0.000003, roughly one quarter that of aluminum. It’s the discrepancy in thermal expansion that would cause structural problems if disparate materials were tightly fastened together.

An immediately striking characteristic of the Wyly Theatre is the elaborate system of ramps and steps leading down to the lower-level entrance to the building. We’ll look at them from a mathematical point of view. Building codes require that each step in a series of stairs have the same height, and the same depth (horizontal distance from the edge of one tread to the edge of the next). That means that with properly constructed stairs, there is always a linear relationship between the distance you move horizontally and the distance you move up or down. That linear relationship is measured with what’s called the *pitch* of the stairs (or ramp), which is simply the vertical distance traveled expressed as a percentage of the horizontal distance traveled. (In mathematics, this quantity is exactly what’s called the *slope* of a line, although we don’t usually write it as a percentage in that context.) So let’s try to figure out the pitches of the various features of this entranceway.

The easiest to compute is the pitch of the shallow staircase to the right of the building. You can simply take a tape measure to see that each tread is 6 inches high and 50 inches deep. So the pitch of that staircase is 6/50, or 12%. The steeper ramps must have the same pitch, because you can just see next to the staircase that the surface of the ramp is parallel to a long stick laid on the top edges of the treads — and parallel lines always have the same slope, or pitch.

But what about the zig-zaggy more shallow ramp down the middle? What is its pitch? Well, we can measure the maximum height of the low walls which separate one switchback of that ramp from the next; it’s about 32 inches. And we can measure the minimum height of the wall at the other end, it’s about 2 inches. So the ramp drops about 30 inches in one full switchback. Now we need to measure the length of one switchback. Do we measure along the ramp? No, because that’s not the horizontal distance; the ramp is slightly angled. We need to find something horizontal to measure along. Fortunately, there’s a line of re-bar holes on every one of these walls that we can use as a guide to horizontal measuring, and you will find that the horizontal length of each ramp is about 30 feet, give or take. So is the pitch of this ramp 30 inches over 360 inches? No, because as the diagram below shows, one full switchback is two ramps, each 30 feet long, that each drop the same amount. So each one drops 30/2 = 15 inches, over 360 inches, for a roughly 4.2% pitch on these ramps.I think it’s now important to ask why the architects bothered to build this extra zig-zag ramp in the middle. The answer is for handicapped accessibility, and so we see that mathematics is being used to understand and plan for the needs of all to be able to access this civic building. The Texas Accessibility Code specifies a maximum pitch that’s safe for those who need to move about in wheelchairs.

(You might at this point ask what that maximum is. Based on the measurements just taken, it’s quite possible you will get suggestions of four or five percent pitch. However, it turns out that the code maximum is 8% pitch. This could spark further discussion as to why, if the maximum was 8%, would the architects have planned such a long ramp only about half the allowed pitch? I don’t know the true answer, but some possibilities that have been suggested are to make the theatre even more welcoming to those in wheelchairs, or because this way the central shallow ramp system divides the entire entrance area into roughly equal thirds, creating a pleasing symmetry to the design. Symmetry is another mathematical concept with strong links to aesthetics and the arts, because most people find symmetry to be inherently attractive.)

There are a couple of other footnotes that we might add here. The shallow ramp travels in a zig-zag fashion. Why didn’t the architect make it straight? The answer is pretty clear; with the required low pitch, if that ramp were straight, it would have to start across the street at the opera house to make it down to the entrance level of the theatre — not a very practical arrangement. What’s interesting here is that a process of folding the ramp up into a zig-zag allowed a longer ramp to fit in a shorter space. Ways that structures can be folded in two and three dimensions are another subject of mathematical study — there is an entire area of mathematical origami — and one with significant practical uses. There are many cases where a large structure must be fit into a small space in a way that will unfold reliably and easily. You can have participants think of such examples; a couple are airbags into your steering wheel or solar collectors for a satellite into the rocket payload bay.

If you’re looking for additional measurements to take, you can examine the pitch of the sidewalk from the street corner up to the side of the building. Is it handicapped-friendly? (It is, but I will leave the measurements that determine its pitch up to you; some creativity is required to get the rise and the horizontal distance for this ramp.)

]]>At this location, you can ask your participants to imagine that they were working on construction for this building, and that the panes of glass in the main grid of windows on this face of the building needed to be ordered. How many panes would you need to order? In the discussion, bring out the fact that there are terraces on the upper levels, so they don’t need that same pane of glass; restrict your count to just the panes that are right on the surface of the building, amidst (the two sections of) the concrete grid.

Some problem-solving strategies that may come up: divide the count into the two sections of the concrete grid. In the right-hand section, the glass panes form a perfect rectangle, so you can count them by multiplying 17 windows across horizontally by 11 windows high to get 187 panes of glass. We will have to add this to the result from the other section of the building. That one is not a rectangle, so we could either divide it into a 4-by-7 rectangle (on the bottom) plus a 3-by-4 rectangle (on the top), or visualize it as a 4-by-11 rectangle with the four top-left panes missing. Either way you work, you will get 40 panes in this section, for a total of 227 panes in all. This exercise provides a good opportunity to illustrate how there can be more than one way to attack a problem, and each different way can give further insight into the nature of the problem.

You can also discuss, time permitting, whether having made the count, you would order exactly 227 panes. Presumably not, because it would be too risky; absolutely nothing could go wrong with any of the panes. Since glass can be fragile, you would likely order some extra. Depending on budgets, you might order as much as 10% extra. How many would that be? Well, 10% of 227 rounds to 23, and adding that to 227 gives a nice round 250 panes of glass as a conservative order. (Note that you will likely be spending money on glass that you will never use, but that’s the price of avoiding the risk of ending up short a pane of glass in case of breakage.)

There’s one other interesting story about this building. In a first draft math tour, the tour leader Glen Whitney asked what was the shape the window panels counted in the challenge above. “Well, they look like squares.” And then he asked what the shape of the large structure projecting out from the building near the top was. “That looks like a square, too.” Glen then stated that they couldn’t *both* be squares. Why? Well, if you count window units across, the big structure is clearly six units across. And counting window units up the side of the big structure, it appears to be five units high. If one rectangle is five-by-six in terms of another rectangle, they clearly can’t both be squares; one or the other might be, or neither, but they can’t both be. However, one of the tour participants quietly pointed out to Glen later in the tour, “If you look at the building from far away, I think you will see that the big structure extends past the top of the building and both it and the windows *are* squares.” Judging from the picture below, I believe that’s absolutely correct. The structure does extend past the top of the building, so that it appears to be exactly six units high, so that it and the individual windows are squares.

I didn’t include this story here with any intention that it would be mentioned on a typical tour. Instead, I think it holds several important lessons for the tour process as a whole. (1) You *will* make mistakes. Acknowledge that. Try your absolute best to minimize those mistakes as close to zero as possible, but realize that no matter how hard you try, you will make a mistake now and then. So don’t present yourself as an infallible authority. (2) Math tours, like all mathematical experiences, are works in progress; they can always be refined, extended, polished, improved, corrected, or expanded. (3) The folks that come along on the tours are a valuable resource. Engage them. Listen to them. And most important of all, be open to learning from them. If you are, you certainly will, and these interactions will enrich your experience as a tour leader and make leading math tours more rewarding.