Geometry

Cutting Up Space Using n Planes

As the capstone of this series on counting, lets look at something that’s a little harder to count by drawing: What is the maximum number of regions into which all of 3-dimensional space can be divided by n planes? We’ll look at two significantly different perspectives. Working up through the dimensions The first question is …

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Overlapping Square Tiles

(A new question of the week) An interesting geometry question came to us in July, about the area of overlap between two squares. The discussion was not long, but leads to some interesting ideas. Here is the initial question, from Vignesh: This is the problem I need help with: Two square tiles of area 9 …

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Vector Basics: Adding Arrows

Because we have had a number of questions about vectors recently, I thought it might be time to look at various facets of that topic. Here, we will start with some ideas about what vectors, and their most basic operations, are. Next week, we’ll get into the far more interesting topic of multiplying vectors. What …

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How Much Does the Earth Curve?

We’ve looked at how to find the circumference of the earth, and how far we can see over the horizon. Another kind of question we’ve had about the curvature of the earth is, how much does it curve over a given distance? That has been asked in several different ways, which lead to some intriguing …

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How Far Can I See?

We have been looking at questions about the roundness of the earth, starting with the general fact, and then the determination of the size of the earth. A very common question is about how that roundness affects what we can see, sometimes as a challenge (“If I can see this, then how can the earth …

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Eratosthenes Measures the Earth

Last time we looked at a couple questions about proving the earth is round, which led into questions about how Eratosthenes measured the earth (though that in itself did not prove the earth is not flat). Let’s look at two questions about that project itself. Eratosthenes for third graders The first is from 1995, and …

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