Category Archives: geometry

Challenge: area of a parallelogram

Posted on June 24, 2021 by Brent

And now for something completely different!1 Suppose we have a parallelogram with one corner at the origin, and two adjacent corners at coordinates and . What is the area of the parallelogram? There are probably many different ways to derive … Continue reading →

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The First Six Books of the Elements of Euclid, by Oliver Byrne (Taschen)

Posted on January 11, 2020 by Brent

Recently for my birthday I received a copy of Oliver Byrne’s 1847 edition of Euclid’s Elements (pictured at right), republished by Taschen Books in 2010. I’ve only just started reading it, but it’s beautiful and fascinating. Oliver Byrne was a … Continue reading →

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SMT solutions

Posted on July 30, 2018 by Brent

In my last post I described the general approach I used to draw orthogons using an SMT solver, but left some of the details as exercises. In this post I’ll explain the solutions I came up with. Forbidding touching edges … Continue reading →

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Drawing orthogons with an SMT solver

Posted on July 12, 2018 by Brent

I’m long overdue to finish up my post series on orthogons as promised. First, a quick recap: An orthogon is a polygon with only right angles. Two orthogons are considered the same if you can turn one into the other … Continue reading →

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Chromatic number of the plane roundup

Posted on June 28, 2018 by Brent

I’ve had fun writing about the Hadwiger-Nelson problem to determine the chromatic number of the plane, but I think this will be my last post on the topic for now! More 7-colorings Of course, the original point of the hexagonal … Continue reading →

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Some words on PWW #22

Posted on June 19, 2018 by Brent

There are lots of patterns to be found in the picture from my previous post! This is a really remarkable tiling. Here are a few special properties I know of: First of all, I hope you realized that the pattern … Continue reading →

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Post without words #22

Posted on June 9, 2018 by Brent
Posted in geometry, pattern | Tagged , , , , , | 13 Comments

The chromatic number of the plane, part 4: an upper bound

Posted on June 2, 2018 by Brent

In my previous posts I explained lower bounds for the Hadwiger-Nelson problem: we know that the chromatic number of the plane is at least 5 because there exist unit distance graphs which we know need at least 5 colors. Someday, … Continue reading →

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The chromatic number of the plane, part 3: a new lower bound

Posted on May 15, 2018 by Brent

In my previous post I explained how we know that the chromatic number of the plane is at least 4. If we can construct a unit distance graph (a graph whose edges all have length ) which needs at least … Continue reading →

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The chromatic number of the plane, part 2: lower bounds

Posted on May 1, 2018 by Brent

In a previous post I explained the Hadwiger-Nelson problem—to determine the chromatic number of the plane—and I claimed that we now know the answer is either 5, 6, or 7. In the following few posts I want to explain how … Continue reading →

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