What we're about

Math Counts is a meetup to engage all things mathematical in a relaxed setting on the fourth Saturday of each month. We strive to make each event accessible to those with rudimentary math skills while also engaging those with more advanced knowledge, so join us no matter what your level of mathematical ability.
Math Counts brings together math aficionados, amateur and professional mathematicians and educators to engage all things mathematical. Our meeting topics range from the elementary to the profound, the practical to the philosophical, and the simple to the complex. Whether we are discussing books or on-line videos, hanging out to discuss recent mathematics news, enjoying mathematics activities, or otherwise imbibing the mathematical, we invite you to join us in a relaxing setting for stimulating polite conversations and activities to participate in the fabric of Philadelphia's vibrant Mathematics tapestry.
Mathematics is surprising, playful, stimulating and profoundly applicable to most aspects of life. Keith Devlin and others call it the science of patterns. Here are some quotes about the subject:

"If the modern world stands on a mathematical foundation, it behooves every thoughtful, educated person to attempt to gain some familiarity with the world of mathematics. Not only with some particular subject, but with the culture of mathematics, with the manner in which mathematicians think and the manner in which they see this world of their own creation."
--- William Byers
"A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street." --- An Old French Mathematician quoted by David Hilbert
"But it should always be insisted that a mathematical subject is not to be considered exhausted until it has become intuitively evident."
--- Felix Klein
"I assert only that in every particular Nature-study, only so much real science can be encountered as there is mathematics to be found in it"
--- Immanuel Kant
"The greatest challenge today, not just in cell biology and ecology but in all of science, is the accurate and complete description of complex systems. Scientists have broken down many kinds of systems. They think they know most of the elements and forces. The next task is to reassemble them, at least in mathematical models that capture the key properties of the entire ensembles."
--- E. O. Wilson, Consilience, p.85.

Upcoming events (1)

Extrema without calculus: Right Prisms, Double Pyramids, Double Cones

Needs a location

We will explore finding minimal surface areas using the theorem of the means. The "theorem of the means" (also called the "theorem of the arithmetic and geometric means") states that the geometric mean of n positive numbers is less than the arithmetic mean of those numbers unless all n numbers are identical.

Although it may seem surprising, this result can be applied to many extrema problems without using calculus. The method is developed in section 5 and 6 of Chapter 8 in George Pólya's "Mathematics and Plausible Reasoning: Induction and Analogy in Mathematics". Chapter 8 in the book stands on its own and can be read independently of the rest of the book.

In particular, this group exploration will examine Chapter 8 §5–6 on pages 121–131 of Pólya's "Induction and Analogy in Mathematics".

Prerequisite: high school geometry and algebra. Familiarity with the theorem of the means as explained in depth in Chapter 8 of Pólya's "Induction and Analogy in Mathematics".

To guide your preparation & participation, consider:

➀ Our first concern will be a casual exploration of the text of Chapter 8 on pages 121–131 in "Induction and Analogy in Mathematics". Bring your notes, questions, insights, difficulties, and critiques for these 11 pages of text.

● Do you have any questions or comments about the text?

● Do you understand the method for using the theorem of the means to find extrema without calculus discussed in §5 and §6?

➁ Our second concern will be an in depth look at #33,36-39 in "Examples and Comments on Chapter VIII" on pages 137–139 in George Pólya's "Induction and Analogy in Mathematics".

• Understanding #34 and 35 may be useful for #36-39. We may review #34 and 35 a bit at the beginning. If you are struggling with the idea of applying the theorem of the means, study my rash and mindless attempts in the one page PDF of my solution to #34.

• These problems invite you to exercise your knowledge and skill to resolve mathematical questions. Bring your partial solutions so we can work together to strengthen everyone's understanding. Through this practice, we will all come to understand the particular situations and the pattern of partial variation better. Even when none of us can solve all the problems, hopefully each of us will understand enough to fill in each others' gaps to give everyone a better idea about how each one works.

Pólya's book is in the public domain, so you can find free copies of it in PDF, EPUB, Kindle, text, and other formats at https://archive.org/details/Induction_And_Analogy_In_Mathematics_1_ (scroll down to the "Download Options"). This PDF might give the best printed copy: https://archive.org/download/Induction_And_Analogy_In_Mathematics_1_/Induction_And_Analogy_In_Mathematics_1_.pdf

In addition to the main material described above, the following passages introduce Pólya's approach in the book which will be useful but not strictly necessary for the event:

The Preface on pages v–x; pay particular attention to §1–4.

The very important Hints to the Reader on pages xi–xii.

Chapter 1 on pages 3–11. Pay special attention to Example Problems 9–14 on pages 9–11.

Photos (30)