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Category Archives: geometry
Challenge: area of a parallelogram
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
The First Six Books of the Elements of Euclid, by Oliver Byrne (Taschen)
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
SMT solutions
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
Posted in computation, geometry
Tagged constraint, crossing, orthogonal, orthogons, perimeter, polygon, segments, SMT, solver
Comments Off on SMT solutions
Drawing orthogons with an SMT solver
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
Posted in computation, geometry
Tagged constraint, drawing, global, local, orthobraces, orthogonal, orthogons, polygon, SMT, solver
1 Comment
Chromatic number of the plane roundup
I’ve had fun writing about the HadwigerNelson problem to determine the chromatic number of the plane, but I think this will be my last post on the topic for now! More 7colorings Of course, the original point of the hexagonal … Continue reading
Some words on PWW #22
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
The chromatic number of the plane, part 4: an upper bound
In my previous posts I explained lower bounds for the HadwigerNelson 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
The chromatic number of the plane, part 3: a new lower bound
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
The chromatic number of the plane, part 2: lower bounds
In a previous post I explained the HadwigerNelson 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