Ratio boxes
Python for Bioinformatics - adventures in bioinformatics
by telliott99
4M ago
I worked up a short new chapter for my Geometry book. It's about a device I'm calling ratio boxes, for want of a better word. When we have similar triangles, we have equal ratios of sides. An example: Above we have three similar right triangles, so we write down the sides in order from smallest to largest, and then repeat, going through each triangle in order. The trick is that any four entries making a rectangle are a valid ratio from this data. In particular, I'm hoping you may be able to see a quick proof of Pythagoras's Theorem. There are several more examples. The most complicated is one ..read more
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Newton's series for the exponential
Python for Bioinformatics - adventures in bioinformatics
by telliott99
7M ago
I've been reading the first chapter of Dunham's Calculus Gallery again (available here). It starts with a discussion of Newton realizing that the binomial theorem (a+b)^n also applies for rational r as in (a+b)^r. In that case, the series does not terminate but is infinite. There is a deep discussion of how he came to that by Dennis and Addington, here. Many great things come out of that including series for the logarithm and inverse sine and several series for π. Dunham also illustrates how Newton inverted or reversed series, for example to turn the inverse sine into the sine and the logarith ..read more
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Broken chord
Python for Bioinformatics - adventures in bioinformatics
by telliott99
8M ago
The theorem of the "broken chord" is ascribed to Archimedes, although his original work has been lost. It was analyzed by the Arabic mathematician Al Biruni in his Book on the Derivation of Chords in a Circle. Here is the general setup: Let A and C be any two points on a circle, and let M be equidistant from both so that arc AM is equal to arc MC. Let B be another point on the circle, lying between A and M, so that AB Drop the perpendicular from M to F on BC. We claim that AB + BF = FC. I will not spoil the fun by giving the proofs here. But these are seven constructions I know about. Draw E ..read more
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Killing geometry
Python for Bioinformatics - adventures in bioinformatics
by telliott99
10M ago
I think it's fair to say that math is not my granddaughter's favorite subject. The whole debate about whether some people are inherently good at math and some are not, is for another day. It is probably relevant that she is home-schooled and using online materials to learn. I've been excited because she's starting geometry, and I really like the subject. So I am presented with this (I'm reconstructing) as the first problem. There are so many things wrong with this that it's hard to know where to start. The biggest one is that she has not previously seen a problem like this being solved. The i ..read more
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Acheson's Geometry
Python for Bioinformatics - adventures in bioinformatics
by telliott99
10M ago
One of my favorite books is David Acheson's The Wonder Book of Geometry (e.g. here). And one of my favorite parts of the book is where he develops a proof that similar right triangles have equal ratios of sides. Here is how I might present it in a slightly more long-winded way. Draw a rectangle (any rectangle) and then draw one diagonal. This forms two right triangles which are congruent (by every method: SSS, SAS or even ASA). Any rectangle is divided by its diagonal into equal areas above and below the line. Next, introduce a point on the diagonal and draw two more rectangles and divide eac ..read more
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Geopandas and maps
Python for Bioinformatics - adventures in bioinformatics
by telliott99
10M ago
Recently I've been exploring maps again, using GeoPandas in Python. I found it confusing at first, but that was mainly because I didn't understand the underlying technology very well, especially Pandas and the shapely geometry library. I've had to brush up on matplotlib, as well. I use Homebrew to obtain my own Python3, rather than relying on Apple's build that is provided with macOS. Some people dislike Homebrew, but I'm not one of them. I never have any trouble with my (simple) "stack", and if I did, I would use a virtual environment. One thing that has changed over time is the need to use ..read more
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Comments
Python for Bioinformatics - adventures in bioinformatics
by telliott99
2y ago
I've had to turn comments off for the blog. Nothing but spam anymore. The intrepid reader will be able to find me. Hint: +"9" and I use gmail ..read more
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Sum of angles: Ptolemy
Python for Bioinformatics - adventures in bioinformatics
by telliott99
2y ago
 The "sum of angles" theorems are incredibly useful in calculus.  These are formulas for the sine (or cosine) of the sum (or difference) of two angles.  There are a number of derivations, of which my favorite is a proof-without-words (here). Recently, I got interested in Ptolemy's theorem, which is illustrated in this graphic: There are proofs based on similar triangles (a bit complicated), and one based on areas with a wonderful trick in the middle.  Another terrific proof-without-words for that is here. It turns out that there are easy proofs for the trigonometric theo ..read more
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Calculating continued fractions
Python for Bioinformatics - adventures in bioinformatics
by telliott99
2y ago
 I've been playing with continued fractions.  Here is undoubtedly the simplest example. And here is sqrt(3) for another.  I find the square root symbol to be a bit of a distraction, so I substituted x for sqrt(3). The standard notation for this is [1;(1,2)].  Here is the derivation for this example: At this point, the trick is to add and subtract 1 on the right-hand side. (In other examples, we turn 1/x - something into 1/x + something at this step).  But for this one, we are done because x - 1 appears on both sides. Then to evaluate it we must chop off the c ..read more
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Pentagons, again
Python for Bioinformatics - adventures in bioinformatics
by telliott99
2y ago
 It's been some years, but I took another look at pentagons.  It's fun to see the connection between phi, the "golden ratio", and the internal triangles in a pentagon. First of all. the marked angles on the left are equal by the isosceles triangle theorem.  Then we do some addition:  3 black + 2 magenta = 4 magenta + 1 black.  We conclude that black = mageneta. But now, we see that the chord above the side on the bottom, which looks like it might be parallel, is parallel, by alternate interior angles.  So we have a bunch of rhombi! Then we notice two types ..read more
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