Available Lectures on Mathematical topics
- Cutting Cake for Greedy People
- The Fourth Dimension
- Doughnuts, Dogbones, and Topology
- Infinity: How to count when you run out of toes
- Circles, Rings, and Tractors: Clever Cleaving for Finding Formulas
- Archimedes, Onions, and Paint Cans
- Circles, Pyramids, Cones, and Spheres
- Zeno’s Arrow → The Concept of Limit
- Buffon’s Needle— π from Breadsticks
- Clear Vision in the Fog of Probabilities
- The Fourth Dimension
- Circles, Pyramids, Spheres, and Archimedes
Cutting Cake for Greedy People
Suppose you have a cake and three greedy people. Some like icing; some like the candy flowers; some like the cake part. Can you devise a method of dividing the cake such that all three people get a fair share and know it is fair? Can you devise a method of dividing the cake into three pieces such that all three people get their favorite piece? Is this a talk in psychology or, just maybe, can clear, logical thought prove that everyone can be happy?
Back to TopThe Fourth Dimension
The fourth dimension sounds eerie, mysterious, and exciting; and it is. Untying knots, stealing gold bricks from closed iron safes, and unfolding hypercubes are all part of the journey. We are transported to this abstract domain by a powerful method of creating ideas, namely, thinking insightfully about the world that we know well. A deep understanding of the simple and familiar is the key to exploring the complex and mysterious, and the fourth dimension illustrates that principal magnificently.
Back to TopDoughnuts, Dogbones, and Topology
Most objects in our everyday world are more or less rigid. But with some flexibility in our mind’s eye, we can imagine them as stretchable and elastic. That image gives us a whole new potential reality to explore. In such an unreasonably contortable universe, we discover things that surprise and amuse us, including learning how to turn our trousers inside out with fully removing them.
Back to TopInfinity: How to count when you run out of toes
Infinity is big. For thousands of years, people also thought it was incomprehensible—an idea so vast that understanding it was beyond the scope of people's finite minds. But a child's method of sharing—'one for me, one for you'—, an Infinite Inn, a barrel containing infinitely many Ping-Pong balls, and a game called Dodge Ball combine to take us to infinity. And beyond.
Back to TopCircles, Rings, and Tractors: Clever Cleaving for Finding Formulas
How do we discover the formulas for the areas of objects such as circles and annuli and the volumes of solids such as cones, pyramids, and spheres? In each case, an effective strategy involves dividing the object into small pieces and seeing how the small pieces can be re-assembled to produce an object whose volume or area is easier to compute. Some of these methods were devised thousands of years ago and some of them seem to be relatively new.
Back to TopArchimedes, Onions, and Paint Cans
Archimedes devised an ingenious method using levers to deduce the formula for the volume of a sphere. The method foreshadowed the idea of the integral in that it involved slicing the sphere into thin sections. Integrals provide effective techniques for computing volumes of solids and areas of surfaces. Inspired by onions, we investigate how a solid can be viewed as layers of surfaces. The derivative reveals the relationship between surface area and volume in the example of the sphere. For less familiar solids, the relationship between volume and surface area can be strange indeed. We will construct a container that holds less than four gallons of paint, but whose walls are infinitely large, so they cannot be painted. This paradoxical state of affairs seems ridiculous at first but is easily possible in the abstract world that includes infinitely tall paint cans.
Back to TopCircles, Pyramids, Cones, and Spheres
How do we find formulas for the areas of objects such as circles and the volumes of solids such as cones, pyramids, and spheres? We can deduce each of these formulas by dividing the object into small pieces and seeing how the small pieces can be assembled to produce the whole. The area of a circle, πr ², is a wonderful example of a formula that we may just remember with no real sense of why it’s true. But we can view the circle in a way that shows clearly whence the formula arises. The process involves a neat method of breaking the circle into pieces and reassembling those pieces. This example and others illustrate techniques of computing areas and volumes that were ancient precursors to the modern idea of the integral.
Back to TopZeno’s Arrow → The Concept of Limit
An arrow is shot from a bow and proceeds toward your heart. But you are calm owing to the following argument that constitutes one of the paradoxes of Zeno; namely, you realize that the arrow must first travel half way to your heart, then it must proceed for half the remaining distance, then half the remaining distance, and so on forever. Since there are infinitely many halves to be traversed, you feel secure in your safety—that is, until the arrow strikes. How can the infinite number of halves be reconciled with the experience that the arrow does reach its target? This paradox illustrates the idea of limit, which is involved in both the derivative and the integral. Indeed, both the derivative and the integral entail infinite processes that result in one coherent answer. The notion of limit makes these infinite processes more meaningful and precise.
Back to TopBuffon’s Needle— π from Breadsticks
Calculus finds applications in many corners of the world, so it should come as no surprise that calculus is useful in many branches of mathematics as well. Here we explore an example where calculus is used to compute a surprising result in the area of probability. Let’s consider the following random event. Suppose we have a sheet of lined, notebook paper and a needle whose length is equal to the distance between consecutive lines on the paper. We now randomly drop the needle onto the paper. We notice that some of the time the needle crosses one of the lines, and some of the time the needle does not cross any line. What fraction of the time will the needle cross a line? The surprising answer is 2/π, and that result uses calculus and breadsticks. So this physical experiment gives a method for estimating the value π.
Back to TopClear Vision in the Fog of Probabilities
Making decisions based on statistical and probabilistic inferences can be tricky. Some statistical and probabilistic results seem counterintuitive, and the logic involved in making good decisions based on those results can be correspondingly surprising. This talk will explore various examples of probabilistic and statistical anomalies including the Two-Sons Paradox, Simpson’s Paradox, Benford’s Law, and Newcomb’s Paradox.
Back to TopThe Fourth Dimension
The fourth dimension sounds eerie, mysterious, and exciting; and it is. Untying knots, stealing gold bricks from closed iron safes, and unfolding hypercubes are all part of the journey. We are transported to this abstract domain by a powerful method of creating ideas, namely, thinking insightfully about the world that we know well. A deep understanding of the simple and familiar is the key to exploring the complex and mysterious, and the fourth dimension illustrates that principal magnificently.
Back to TopCircles, Pyramids, Spheres, and Archimedes
How do we discover the formulas for the areas of objects such as circles and triangles and the volumes of solids such as cones, pyramids, and spheres? In each case, an effective strategy involves dividing the object into small pieces and seeing how the small pieces can be re-assembled to produce an object whose volume or area is easier to compute. One of the most impressive of these triumphs occurred in the third century B.C., when Archimedes devised an ingenious method using levers to deduce the formula for the volume of a sphere. All these methods foreshadowed the concept of the integral.
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