# Math Circle

This Math Circle is for middle school (and high school) students who enjoy thinking about mathematical ideas, problems, and puzzles.

For further information contact: Daniel Shapiro.

Info on floor tilings in the OSU Math Tower:  Tessellation Project.

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2024:

•May 5, 2024:  Surfaces by Cutting and Pasting.
Sunday 2:00 – 4:00  in room:  154 Math Tower

Make a cylinder by gluing the two vertical sides of a rectangle.  Then glue the horizontal sides to get a torus.  If we unroll that torus over the whole plane, each point on the torus corresponds to a lattice of points in the plane.
Allowing twists when gluing a rectangle’s edges, we can produce Klein bottles and projective planes. What other smooth surfaces can be made by gluing various edges of a polygon?

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• April 21, 2024:  Geometric Glue.
Sunday 2:00 – 4:00  in room:  154 Math Tower

In geometry, we’re concerned with measuring lengths, angles, areas.  But for this math circle, I prepared a collection of geometric puzzles that depend less on exact angles or lengths, and more on ‘shape’ in general.  You’ll draw curves to avoid other curves, or to cross another curve a certain number of times — but the curves you’ll draw will be on surfaces, like a 2-holed donut!  Since it’s hard to understand curves on these objects, we’ll develop some techniques for making it possible to visualize these puzzles.

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• April 7, 2024:  Traffic Patterns.
Sunday 2:00 – 4:00  in room:  154 Math Tower

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• March 24, 2024:  Spherical Geometry and Beyond.
Sunday 2:00 – 4:00  in room:  154 Math Tower

Can we do geometry on the surface of a sphere?
What could be a line?  How might we measure distances and angles?  In what ways would this geometry be the same or different from Euclidean (flat) geometry?  What other geometries are possible?

You are invited to bring a ball (basket, soccer, volley, etc) to help our explorations.

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• March 10, 2024:  Fractions.
Sunday 2:00 – 4:00  in room:  154 Math Tower

Fractions  m/n  between 0 and 1 and with denominator n < 6 can be listed in order of size:

0/1,   1/5,   1/4,   1/3,   2/5,   1/2,   3/5,   2/3,   3/4,   4/5,   1/1.

By examining lists of this type, we can discover some interesting numerical patterns.  We will try to figure out why those observed patterns always hold true.

These ideas can lead us to “continued fractions” and related topics.  Centuries ago, these ideas were developed to help people compute good approximations to numbers like square-root-of-2.

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• February 25, 2024:  Using Euclid’s Algorithm.
Sunday 2:00 – 4:00  in room:  154 Math Tower

Euclid tells us how to compute the “greatest common divisor” of two integers.  That procedure can be used to solve problems like this:
At a strange local store, each apple costs 28 cents and each orange costs 39 cents.  How many apples and oranges yield a total cost of exactly 10 dollars?  In other words, find positive integers  x, y  satisfying   28x + 39y = 1000 ?

Bring a calculator to this session !

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• February 11, 2024:  Geometric Constructions.
Sunday 2:00 – 4:00  in room:  154 Math Tower

2300 years ago, Greek geometers asked students to construct various figures using only “Euclidean tools”:  a straight-edge and a compass.  With those tools we can construct squares, equilateral triangles, angle bisectors, tangents to circles, etc.   We will use paper, pencils, rulers, and compasses to explain those constructions.
Some famous construction questions remained unsolved for 2000 years.  For instance is it possible to square a circle, to duplicate a cube, or to trisect a given angle ?

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• January 28, 2024:  Hindsight.
Sunday 2:00 – 4:00  in room:  154 Math Tower
Guest leader:  Anna Davis (math professor at Ohio Dominican University)

Can a photograph be used to determine where the camera was located when the picture was taken?  We will use geometric principles, and hands-on experiments to answer this question.  Bring your camera phones!

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• January 14, 2024:  Some Math Problems.
Sunday 2:00 – 4:00  in room:  154 Math Tower

Here’s one classic question:
For what numbers n is it possible to tile a square with  n smaller squares?

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2023:

• December 10, 2023:  Lattice Polygons.
Sunday 2:00 – 4:00  in room:  154 Math Tower

A point (x, y) in the plane is a  lattice point  if both  x  and  y  are integers.  Then lattice points are the corner points of the usual square grid covering the plane.

Suppose  P   is a polygon whose vertices are at lattice-points.  Define numbers I and B :

I  =  number of lattice-points inside  P.
B  =  number of lattice-points on the boundary of  P.

We will discuss the simple formula that connects those numbers with  area( P ).

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• November 26, 2023:  Some Math Problems.
Sunday 2:00 – 4:00  in room:  154 Math Tower

If you know some contest problems that you would like to discuss, please send me details.
Here’s one classic question:
At a party with seven children, the dessert is a square cake with icing on the top and on the four sides.  Is there a simple way to cut the cake into seven pieces so that each piece has the same volume of cake and the same area of icing?
Each child needs a connected piece (not several different pieces, or a pile of mush.)

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• November 12, 2023:  Infinity.
Sunday 2:00 – 4:00  in room:  154 Math Tower

How can we precisely define what is meant by the “number of elements” in a set  S ?  This is especially hard to do when  S  is infinite.  To begin, we define when two sets  S  and  T  have the *same* size.
Can two infinite sets have different sizes?

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• October 29, 2023:  Racetrack offsets.
Sunday 2:00 – 4:00  in room:  154 Math Tower

On an oval racetrack, the runner in the inside lane goes one-quarter mile, starting and ending at the same point.  The next lane is longer but race officials want a single finish line for the race.  Then in that lane, the starting point must be a bit ahead of the inside-lane’s starting point.
– –  How long is that offset? – –
Does it depend on the distance between the two tracks?  On the total length of the race?  On the shape of the racetrack?
We will discuss those questions and generalize in various ways.

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• October 15, 2023:  Some Counting Questions.
Sunday 2:00 – 4:00  in room:  154 Math Tower
Leader = Nimish Shah (Professor, OSU Department of Mathematics).

Suppose each face of a cube is painted one color, and all the face-colors are different.  In how many ways can this be done using six colors?
What are some generalizations of this problem?

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• October 1, 2023:  Complex numbers.
Sunday 2:00 – 4:00  in room:  154 Math Tower

“Real” numbers correspond to points on the usual number line.
The number  i = √-1  (square root of -1) is not a real number.  (Does it even exist?)
“Complex” numbers are expressions like  2 + 3i  and  √3 – 12i.  We will discuss some interesting connections between complex numbers and plane geometry.

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• September 17, 2023:  Areas of triangles.
Sunday 2:00 – 4:00  in room:  154 Math Tower

Given the coordinates of the 3 corners of a triangle, how can you find the area?
What are some ways to generalize this question?

We will also ask students (and parents):
What dates and times are best for future meetings of this Circle?

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• April 29, 2023:  Drawing stars.
Saturday 3:30 – 5:30  in room:  154 Math Tower

When several points are equally spaced around a circle, we may connect them by line segments to draw a star.  Properties of those stars are related to modular arithmetic.

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• April 8, 2023:  Thinking about doodles.
Saturday 3:30 – 5:30  in room:  154 Math Tower

Students will draw some closed curves, allowing some self-intersections. How can properties of those curves be investigated mathematically?

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