Edges and Faces: A Matter of Definition
(An archive question of the week) Having looked at the matter of faces, edges, and vertices from several different perspectives, I want to look at one more question and answer, to tie it all together.
Definitions
Do Curved Surfaces Have Faces, Edges, and Vertices?
Having discussed how to count faces, edges, and vertices of polyhedra, and then looked at Euler’s formula that relates them (not only in polyhedra but in graphs on planes and other surfaces), we need to consider a question we have received at least 100 times: Are these terms even defined (or defined correctly) for cylinders, …
Do Curved Surfaces Have Faces, Edges, and Vertices? Read More »
More on Faces, Edges, and Vertices: The Euler Polyhedral Formula
Last time we looked at how to count the parts of a polyhedron, and a mention was made of Euler’s Formula (also called the Descartes-Euler Polyhedral Formula), which says that for any polyhedron, with V vertices, E edges, and F faces, V – E + F = 2. We should take a close look at that simple, yet amazing, …
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Epsilons, Deltas, and Limits — Oh, My!
Using the epsilon-delta definition of a limit in calculus can be challenging. (That’s why, after using it for a few examples, we derive some easier techniques, and never use the definition directly unless we have to!) We’ll start with an overview of what the definition means, and then look at several examples of how it …
Three Meanings of “Percentile”
(An archive problem of the week) Having just discussed quartiles, I want to look at related issues concerning percentiles. There, I briefly mentioned different perspectives on the concept of quartile, and focused on differences in the details of the calculations; here I will focus mostly on the different perspectives, and then touch on variations in …
The Many Meanings of “Quartile”
Some time ago I discussed various issues pertaining to the concept of median in statistics. The same issues, and more, affect the concept of quartile (the median being the second quartile), so much so that different statistical software packages produce many different answers for quartiles. I have seen this affect students, who are taught one …
What is Multiplication … Really?
I want to close out this series on multiplication with a very different kind of question. We have seen that multiplication of natural numbers can be modeled as a repeated sum of the multiplicand, taken the number of times indicated by the multiplier; and that the terms “multiplier” and “multiplicand” reflect only this model, not …
What is Multiplication? How (Not) to Teach It
Last time we looked at the roles of multiplier and multiplicand from several perspectives. This time, I want to focus on one extended discussion about how children should be taught to think of multiplication.
What is Multiplication? Multiplicand and Multiplier
We have received many questions over the years about the meaning of multiplication. When we multiply , what are we really doing? This can confuse not only students and their parents, but also teachers. The next couple posts will deal with various aspects of this question.
Properties as Axioms or Theorems
To close out this series that started with postulates and theorems in geometry, let’s look at different kinds of facts elsewhere in math. What is commonly called a postulate in geometry is typically an axiom in other fields (or in more modern geometry); but what about those things we call properties (in, say, algebra)?