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 – 2025:
•October 6, 2024: Infinities.
Sunday 2:00 – 4:00 in room: 154 Math Tower
If S is a set, how can we count the number of elements in S ? We want to assign a “cardinality” to S, expressing its number of elements. For instance, the sets {0, 1, 2} and
{a, b, c} each have cardinality 3. But what does “three” mean?
* How can we define “counting numbers” in some meaningful way?
Some sets are finite while others are infinite. But what is the definition of “finite”?
The set N = { 1, 2, 3, . . . } (all the counting numbers) is infinite. Do some infinite sets contain more elements? Would that produce a larger size of infinity?
We will consider these questions and related topics.
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•September 8, 2024: Cutting and Pasting in Geometry.
Sunday 2:00 – 4:00 in room: 154 Math Tower
Polygons A and B are “scissors-congruent” if A can be cut into pieces (using straight cuts) and those pieces can be reassembled to form B. Can you explain why any triangle is scissors-congruent to a rectangle? Is it clear that scissors-congruent polygons must have the same area? The converse question is:
If polygons A and B have equal areas, must they be scissors-congruent ?
We will discuss this question and explore some related geometric ideas.
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PREVIOUS YEARS.
2023 – 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
Leader = Jim Fowler
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
Leader = Michael Meeks
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• March 24, 2024: Spherical Geometry and Beyond.
Sunday 2:00 – 4:00 in room: 154 Math Tower
Leader = Brad Findell
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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