tag:blogger.com,1999:blog-52344362466646883812024-02-08T08:10:20.207-08:00Mathematics, biology and recreationJonathan Dushoffhttp://www.blogger.com/profile/03197609136976213552noreply@blogger.comBlogger10125tag:blogger.com,1999:blog-5234436246664688381.post-35354074858495134762020-03-20T13:04:00.001-07:002020-03-20T13:05:41.244-07:00Sero-surveys<br />
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Sero-surveys</h1>
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You can comment on <a href="https://twitter.com/jd_mathbio/status/1240819955769147392" style="color: #1756a9; text-decoration-line: none;">the tweeted version of this article</a></div>
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It’s time for serious consideration of sero-surveys.</div>
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Lockdown cannot continue forever. Governments and societies will be facing tough choices about COVID soon. These choices should be made with as much information as possible.</div>
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A critical place where there are big gaps in the information puzzle is the infection-fatality rate (IFR). Population-level surveys, done with stratified sampling, have a huge upside potential in gaging the severity of COVID infection.</div>
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It’s important to distinguish between case-fatality and infection-fatality. Case-fatality is a proportion of individuals with identified COVID <em>disease</em>. Infection-fatality is a proportion of all individuals with <em>infection</em>.</div>
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Both disease and infection can be hard to define, but disease is typically much harder to pin down, with definitions and their applications changing with place and time. Both ratios are important, but it is IFR which is critical for planning.</div>
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Unless an effective vaccine is rapidly deployed, the majority of people around the globe will fight off coronavirus infection and develop antibodies within the next few years. We want to know the proportion of <em>everyone</em> that is likely to be suffer serious consequences, or die.</div>
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Now is a critical time. It is only now becoming possible to contemplate large-scale sero-surveys, and there is not much time left before we have to make hard choices about balancing between disease control and vital economic and social concerns.</div>
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Doing these surveys will not be easy. Effort spent gathering information about people infected in the past will inevitably take some attention away from those infected in the present or near future. Even were the long-term benefit more clear, this is not an easy choice. International cooperation will also be important. Places that already have large numbers of cases have more information to provide, but less resources to spare on gathering information for the future.</div>
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Nonetheless, gathering this information could be immeasureably valuable in planning. We need to weigh this value now.</div>
</article>Jonathan Dushoffhttp://www.blogger.com/profile/03197609136976213552noreply@blogger.com0tag:blogger.com,1999:blog-5234436246664688381.post-67347320986648389272017-09-27T16:07:00.000-07:002017-09-27T16:07:33.157-07:001/19<div style="background-color: #fdfdfd; color: #111111; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 16px; margin-bottom: 15px; padding: 0px;">
I was <em>so</em> excited the other night when I realized I had figured out something new and interesting about the decimal expansion of 1/19. Something I hadn’t noticed, despite the fact that 1/19 has been one of my favorite decimal expansions for decades.</div>
<h2 id="powers-of-5" style="background-color: #fdfdfd; color: #111111; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 32px; font-weight: 400; margin: 0px 0px 15px; padding: 0px;">
Powers of 5</h2>
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I’ll start at the beginning. Repeating decimals tend to have all sorts of cool patterns and properties. A nice simple one about 1/19 is that you can generate it from the powers of 5. More or less.</div>
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1/19 is 5/95 = 5/(100-5) ≡ 5/(H-5), where H=100. If we define h=1/H and multiply, then:</div>
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1/19 = 5h/(1-5h) = 5 + 5²h + 5³ h² + …</div>
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We can try to write that out as:</div>
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. 05 25 125 625 3125 … .</div>
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Each block here represents a pair of decimal digits, so it’s clear that something is wrong: 125 and later entries don’t fit. We could arrange them to overlap properly and add:</div>
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. 05 26 31 56 … .</div>
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But that’s pretty unwieldy, and just gets worse and worse.</div>
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The better way is “pre-emptive carrying”. Instead of saying “5×5 = 25”, we say “5×5 + the 1 that we know we’re going to carry in a moment = 26” – and then just carry on as if we always had a 26. Then we multiply 26×5=130, give back the 1 at the beginning, and borrow the 1 that we’re about to generate (when we multiply by 5 the next time) to get 31. Pretty wild, but it always works:</div>
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<pre class="highlight" style="background: rgb(238, 238, 255); border-radius: 3px; border: 1px solid rgb(232, 232, 232); font-size: 15px; margin-bottom: 15px; overflow-x: auto; padding: 8px 12px;"><code style="border-radius: 3px; border: 0px; padding: 1px 0px;">1/19 =
.
05 (×5 + 1 = 26) (the one is pre-emptively carried from 131)
26 (×5 + 1 = 131) (the one is pre-emptively carried from 157)
31 (×5 + 2 = 157) (the two is pre-emptively carried from 289)
57 (×5 + 4 = 289) …
89 (×5 + 2 = 447)
47 (×5 + 1 = 236)
36 (×5 + 4 = 184)
84 (×5 + 1 = 421)
21 (×5 + 0 = 105)
05
…
</code></pre>
</div>
<h3 id="checking" style="background-color: #fdfdfd; color: #111111; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 26px; font-weight: 400; margin: 0px 0px 15px; padding: 0px;">
Checking</h3>
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We can tell it’s right because it repeats with the right period (it has to be 18 or a factor of 18, more on that later (probably not, unless someone asks me)), and because it produces the same answer as the other crazy methods we’re going to try later.</div>
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We can also tell that it’s right using <code class="highlighter-rouge" style="background-color: #eeeeff; border-radius: 3px; border: 1px solid rgb(232, 232, 232); font-size: 15px; padding: 1px 5px;">bc</code></div>
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<pre class="highlight" style="background: rgb(238, 238, 255); border-radius: 3px; border: 1px solid rgb(232, 232, 232); font-size: 15px; margin-bottom: 15px; overflow-x: auto; padding: 8px 12px;"><code style="border-radius: 3px; border: 0px; padding: 1px 0px;">> echo "scale=24; 1/19" | bc -l | perl -pe "s/[0-9]{2}/ $&/g"
. 05 26 31 57 89 47 36 84 21 05 26 31
</code></pre>
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Bizarrely, we <em>can’t</em> tell it’s right using R:</div>
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<pre class="highlight" style="background: rgb(238, 238, 255); border-radius: 3px; border: 1px solid rgb(232, 232, 232); font-size: 15px; margin-bottom: 15px; overflow-x: auto; padding: 8px 12px;"><code style="border-radius: 3px; border: 0px; padding: 1px 0px;">> print(1/19, digits=20)
[1] 0.052631578947368418131
</code></pre>
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R is perfectly willing to give you up to 22 digits, but only seems to get the first 16 right, ever.</div>
<h3 id="another-power-example" style="background-color: #fdfdfd; color: #111111; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 26px; font-weight: 400; margin: 0px 0px 15px; padding: 0px;">
Another power example</h3>
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Another cool example of pre-emptive carrying, and this sort of series, is 1/49. Try it yourself. This code gives a nice-looking version of the answer, for comparison (or for the lazy)</div>
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<pre class="highlight" style="background: rgb(238, 238, 255); border-radius: 3px; border: 1px solid rgb(232, 232, 232); font-size: 15px; margin-bottom: 15px; overflow-x: auto; padding: 8px 12px;"><code style="border-radius: 3px; border: 0px; padding: 1px 0px;">echo "scale=44; 1/49" | bc -l | perl -pe "s/[0-9]{2}/ $&/g"
</code></pre>
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<h2 id="dividing-by-4" style="background-color: #fdfdfd; color: #111111; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 32px; font-weight: 400; margin: 0px 0px 15px; padding: 0px;">
Dividing by 4</h2>
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Anyway, that’s not the cool part. I’ve known that forever, thanks largely to Tim Koch.</div>
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<em>Another</em> nice way to derive the expansion for 1/19 is by expanding 1/19=21/399.</div>
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1/19 = 21/(4H-1) = 21h/(4-h) = 21/4 h + 21/4² h² + …</div>
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It would be a huge mess to do this just by blocks, something like:</div>
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. 0525 013125 00328125 …</div>
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But we can do it neatly with another trick: putting off the remainder. 21/4 = 5, with remainder 1, so we write 05 and then save the 1. For the next step, we put the 1 in front of 05 and 105/4 = 26r1. And so on.</div>
<div class="highlighter-rouge" style="background-color: #fdfdfd; color: #111111; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 16px;">
<pre class="highlight" style="background: rgb(238, 238, 255); border-radius: 3px; border: 1px solid rgb(232, 232, 232); font-size: 15px; margin-bottom: 15px; overflow-x: auto; padding: 8px 12px;"><code style="border-radius: 3px; border: 0px; padding: 1px 0px;">1/19 =
. 21/4 = 5r1
05 105/4 = 26r1
26 126/4 = 31r2 (the 1 is the remainder from the line above)
31 231/4 = 57r3 (the 2 is the remainder from the line above)
57 357/4 = 89r1 …
89 189/4 = 47r1
47 147/4 = 36r3
36 336/4 = 84r0
84 084/4 = 21r0
21 021/4 = 05r1
05
…
</code></pre>
</div>
<h2 id="fibonacci-style" style="background-color: #fdfdfd; color: #111111; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 32px; font-weight: 400; margin: 0px 0px 15px; padding: 0px;">
Fibonacci style</h2>
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Same answer. What are the odds‽</div>
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But <em>still</em> not the part that got me excited recently. I figured that out decades ago as well.</div>
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The reason I’m excited, is that I was recently thinking about <a href="https://groups.google.com/forum/#!search/rec.puzzles$20dushoff$20fibonacci$20repeating$20decimals/rec.puzzles/-0zwc8AS_K4/5U-XavaZZqgJ" style="color: #1756a9; text-decoration-line: none;">this conversation from 1999</a> (of course). It talks about how:</div>
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1/89 = 1/(T² - T - 1) (where T=ten), has an expansion that follows the rules of <a href="https://en.wikipedia.org/wiki/Fibonacci_number" style="color: #1756a9; text-decoration-line: none;">the Fibonacci numbers</a>(each number is based on the sum of the two before it). So that’s cool, but 1/89 is kind of a long expansion, and expanding things in groups of 1 (instead of 2) decimals feels a bit ticky. By the way, there’s also a one-digit way to do 1/19: since 19=2T-1, you can expand it by dividing each digit by two (exactly parallel to dividing pairs of digits by 4).</div>
<div style="background-color: #fdfdfd; color: #111111; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 16px; margin-bottom: 15px; padding: 0px;">
So the other day, in the shower, I was wondering about 9899 = H² - H -1, and whether it has any nice factors. Imagine my surprise when I found out that 1/19 = 521/9899! That means that we can write the expansion 1/19 using Fibonacci rules.</div>
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<pre class="highlight" style="background: rgb(238, 238, 255); border-radius: 3px; border: 1px solid rgb(232, 232, 232); font-size: 15px; margin-bottom: 15px; overflow-x: auto; padding: 8px 12px;"><code style="border-radius: 3px; border: 0px; padding: 1px 0px;">1/19 =
. start with the 5 from 521
05 (+ 21 = 26) (the 21 is from 521)
26 (+ 5 = 31)
31 (+ 26 = 57)
57 (+ 31 + 1 = 89) (the one is pre-emptively carried from the next sum)
89 (+ 57 + 1 = 147) …
47 (+ 89 = 136)
36 (+ 47 + 1 = 84)
84 (+ 36 + 1 = 121)
21 (+ 84 = 105)
05
…
</code></pre>
</div>
<div style="background-color: #fdfdfd; color: #111111; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 16px; margin-bottom: 15px; padding: 0px;">
Yet another fun way to generate the same decimal! I can die happy now.</div>
Jonathan Dushoffhttp://www.blogger.com/profile/03197609136976213552noreply@blogger.com0tag:blogger.com,1999:blog-5234436246664688381.post-56083453680739745752017-01-30T16:19:00.003-08:002017-01-31T11:59:16.554-08:00Pythagorean triples plot<script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript">
</script>
<br />
Talking with David Earn and Ben Bolker about simple, mathy programs that make nice pictures (for pedagogical reasons).<br />
<br />
David suggested plotting Pythagorean triples. He said to reduce them to rationals, but I couldn’t figure out a good way to do that (the natural way just produces points on a circle), so I stuck with the integers.<br />
<br />
<a href="http://dushoff.github.io/notebook/pythagoras.R">This code</a> plots (a, b) from Pythagorean triples <span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mi>c</mi><mn>2</mn></msup></math>" id="MathJax-Element-1-Frame" role="presentation" style="border: 0px; direction: ltr; display: inline; float: none; font-size: 16px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-1" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 5.532em;"><span style="border: 0px; display: inline-block; font-size: 16.64px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 5.291em;"><span style="border: 0px; clip: rect(1.205em 1005.29em 2.467em -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.221em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-2" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msubsup" id="MathJax-Span-3" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.965em;"><span style="border: 0px; clip: rect(3.428em 1000.48em 4.21em -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.023em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-4" style="border: 0px; display: inline; font-family: "mathjax_math"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">a</span><span style="border: 0px; display: inline-block; height: 4.029em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.544em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.384em; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-5" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 11.7645px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 4.029em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-6" style="border: 0px; display: inline; font-family: "mathjax_main"; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.243em; position: static; transition: none; vertical-align: 0px;">+</span><span class="msubsup" id="MathJax-Span-7" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.243em; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.844em;"><span style="border: 0px; clip: rect(3.128em 1000.42em 4.21em -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.023em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-8" style="border: 0px; display: inline; font-family: "mathjax_math"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">b</span><span style="border: 0px; display: inline-block; height: 4.029em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.424em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.384em; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-9" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 11.7645px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 4.029em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-10" style="border: 0px; display: inline; font-family: "mathjax_main"; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.303em; position: static; transition: none; vertical-align: 0px;">=</span><span class="msubsup" id="MathJax-Span-11" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.303em; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 0.844em;"><span style="border: 0px; clip: rect(3.428em 1000.42em 4.21em -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.023em; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-12" style="border: 0px; display: inline; font-family: "mathjax_math"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">c</span><span style="border: 0px; display: inline-block; height: 4.029em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; left: 0.424em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.384em; transition: none; vertical-align: 0px;"><span class="mn" id="MathJax-Span-13" style="border: 0px; display: inline; font-family: "mathjax_main"; font-size: 11.7645px; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">2</span><span style="border: 0px; display: inline-block; height: 4.029em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 2.227em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.066em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.122em; width: 0px;"></span></span></nobr></span>, for <span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>&#x2264;</mo><mn>50</mn></math>" id="MathJax-Element-2-Frame" role="presentation" style="border: 0px; direction: ltr; display: inline; float: none; font-size: 16px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-14" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 2.888em;"><span style="border: 0px; display: inline-block; font-size: 16.64px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 2.767em;"><span style="border: 0px; clip: rect(1.385em 1002.71em 2.527em -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -2.221em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-15" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-16" style="border: 0px; display: inline; font-family: "mathjax_math"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">c</span><span class="mo" id="MathJax-Span-17" style="border: 0px; display: inline; font-family: "mathjax_main"; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.303em; position: static; transition: none; vertical-align: 0px;">≤</span><span class="mn" id="MathJax-Span-18" style="border: 0px; display: inline; font-family: "mathjax_main"; line-height: normal; margin: 0px; padding: 0px 0px 0px 0.303em; position: static; transition: none; vertical-align: 0px;">50</span></span><span style="border: 0px; display: inline-block; height: 2.227em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 0.941em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.184em; width: 0px;"></span></span></nobr></span>. The circles get bigger at rate <span class="MathJax" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mi>c</mi></msqrt></math>" id="MathJax-Element-3-Frame" role="presentation" style="border: 0px; direction: ltr; display: inline; float: none; font-size: 16px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; position: relative; white-space: nowrap; word-spacing: normal; word-wrap: normal;" tabindex="0"><nobr aria-hidden="true" style="border: 0px; line-height: normal; margin: 0px; max-height: none; max-width: none; min-height: 0px; min-width: 0px; padding: 0px; transition: none; vertical-align: 0px;"><span class="math" id="MathJax-Span-19" style="border: 0px; display: inline-block; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 1.325em;"><span style="border: 0px; display: inline-block; font-size: 16.64px; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.265em;"><span style="border: 0px; clip: rect(0.123em 1001.26em 1.505em -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.019em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-20" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="msqrt" id="MathJax-Span-21" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span style="border: 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; padding: 0px; position: relative; transition: none; vertical-align: 0px; width: 1.265em;"><span style="border: 0px; clip: rect(3.428em 1000.42em 4.21em -999.997em); left: 0.844em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -4.023em; transition: none; vertical-align: 0px;"><span class="mrow" id="MathJax-Span-22" style="border: 0px; display: inline; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;"><span class="mi" id="MathJax-Span-23" style="border: 0px; display: inline; font-family: "mathjax_math"; font-style: italic; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">c</span></span><span style="border: 0px; display: inline-block; height: 4.029em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(0.965em 1000.42em 1.325em -999.997em); left: 0.844em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -1.86em; transition: none; vertical-align: 0px;"><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: initial; border-right-style: initial; border-top-style: solid; border-width: 1.3px 0px 0px; display: inline-block; height: 0px; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.057em; width: 0.424em;"></span><span style="border: 0px; display: inline-block; height: 1.085em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span><span style="border: 0px; clip: rect(3.068em 1000.84em 4.39em -999.997em); left: 0em; line-height: normal; margin: 0px; padding: 0px; position: absolute; top: -3.963em; transition: none; vertical-align: 0px;"><span style="border: 0px; font-family: "mathjax_main"; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px;">√</span><span style="border: 0px; display: inline-block; height: 4.029em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span></span></span><span style="border: 0px; display: inline-block; height: 1.025em; line-height: normal; margin: 0px; padding: 0px; position: static; transition: none; vertical-align: 0px; width: 0px;"></span></span></span><span style="border-bottom-style: initial; border-color: initial; border-image: initial; border-left-style: solid; border-right-style: initial; border-top-style: initial; border-width: 0px; display: inline-block; height: 1.191em; line-height: normal; margin: 0px; overflow: hidden; padding: 0px; position: static; transition: none; vertical-align: -0.372em; width: 0px;"></span></span></nobr></span> (chosen arbitrarily and kind of looks nice). The box goes from (-50, 50) on each axis. It has some nice patterns.<br />
<img alt="Plot of points from Pythagorean triples" src="https://dushoff.github.io/notebook/git_push/pythagoras.Rout.png" /><br />
<a href="http://dushoff.github.io/notebook/git_push/pythagoras.Rout.pdf">pdf version. (Looks nice, embiggens nicely.</a>)<br />
<br />
There are two natural ways to interpret these points: as pairs of integers, or (cooler) as complex integers (see also <a href="http://lalashan.mcmaster.ca/theobio/math/index.php/Complex_factoring">my post about factoring in the complex plane</a>). There are three natural ways to think of the points that are not on the axes, and three corresponding ways to decide which points to plot on the axes:<br />
<ul>
<li>If we think of the points as representing <a href="https://en.wikipedia.org/wiki/Pythagorean_triple">Pythagorean triples</a>, in the classic sense, the axes should have no points. This is the first way I made the plot, because that was what was suggested.</li>
<li>If we think of the points as representing points on the integer lattice that are an integral distance from the origin, the points off the axes remain unchanged, but we should put a point at each lattice point on the axes.</li>
<li>Finally, if we think of the points as representing square numbers in the complex plane, we should put points at square locations on one of the axes (conventionally, the horizontal axis), but not the other.</li>
</ul>
Now that I write it down, I realize I didn’t actually do any of those things! Sorry! Feel free to tweak the code, but this way looks nice. I simply extended a standard Pythagorean triple definition to <em>not</em> exclude 0 (I kind of thought that matched way 3, but now it doesn’t.)<br />
<br />
Based partly on suggestions from Ethan Bolker, I colored the points:<br />
<ul>
<li>black for “imprimitive” triples (ones which can be generated as integer multiples of other triples)</li>
<li>blue for primitive triples which can be factored into other triples <em>in the complex plane</em></li>
<li>red for triples which are prime <em>even in the complex plane</em> (ie., they represent complex integers which are squares of primes)</li>
</ul>
Now arise more questions about how to color points on the axes. But I have to get this post off my desk before I go crazy!Jonathan Dushoffhttp://www.blogger.com/profile/03197609136976213552noreply@blogger.com0tag:blogger.com,1999:blog-5234436246664688381.post-71447613368930726812015-03-04T13:12:00.000-08:002015-03-04T13:12:13.185-08:00<i>Posted this on reddit, but I don't think it got "red"</i><br />
<br />
Man. My blog is only ever read by people I know, and I certainly would have been more careful if I'd had any idea this was going to blow up.
Friends have suggested I should clarify some points:<br />
<br />
* <i>Nobody</i> suffered permanent marks on their transcript or lost marks from this<br />
<br />
* <i>One</i> person lost a bunch of time retaking the course, the others had to go through an ordeal being grilled by me and were then told that I had decided not to follow up.<br />
<br />
* I did not enjoy the grilling. Didn't really enjoy any of it <i>after</i> the amazing seating discovery (which may be one of the reasons it took me many years to post).<br />
<br />
* It is not at all clear that three or four of these people were innocent: on the occasions where we've observed cheating in person, it has been collaborative.<br />
<br />
* I followed up according to my understanding of the McMaster rules. If I find evidence of dishonesty, I'm expected to report it, investigate it, and then report my findings.<br />
<br />
The reason the part about grilling them was written cavalierly is that, until now, I've mostly been criticized (though lightly) for letting 7 of the 8 off scot free. Had I expected this to reach a wide audience, I would have written more carefully.<br />
<br />
What else? In this test, I think people were broken into big blocks by last name, but were then free to sit where they wanted. The questions were MC questions (often tricky), and I wouldn't expect a large fraction of common wrong answers from people who studied together. It's hard to be sure, though, which is one of the reasons I looked for outliers, instead of using statistics, and then verified with the seating chart. I honestly hadn't thought about the correlation between sitting together and studying together, but if any of the eight people had offered that as a defense, I would have.Jonathan Dushoffhttp://www.blogger.com/profile/03197609136976213552noreply@blogger.com5tag:blogger.com,1999:blog-5234436246664688381.post-1403882311995442832015-02-17T13:07:00.001-08:002015-02-17T13:07:03.336-08:00Finding cheaters using multiple-choice comparisons<h2>
<span class="mw-headline" id="Summary">Summary</span></h2>
An interesting method by which I found out that people were cheating on my final exam.<br />
<h2>
<span class="mw-headline" id="Background">Background</span></h2>
I use different versions of midterm examinations to discourage
cheating in my population biology class (~200 students). When the course
started, I used to do the same thing for the final exam, but it was a
little more complicated, because the final exam is administered by the
registrar's office, not by me and my teaching team.<br />
At some point, somebody advised me not to bother with versions: the
registrar's office is supposed to be professional about administration,
and they <i>usually</i> mix people who are taking different exams in the
same room, so I stopped bothering with different versions for the final
exam for a year or two. I do it again now, and you'll see why.<br />
<h2>
<span class="mw-headline" id="The_incident">The incident</span></h2>
In the year in question, my exam was given in two separate
medium-sized rooms. My class was alone in these two rooms. I received a
report from the invigilators in Room 1 about suspicious behaviour. They
had warned a couple of students for acting strangely, and then warned
them again. They weren't prepared to say that they were sure that the
students were cheating, but wanted me to compare their answer slates. In
retrospect, they should have left the students alone until they were
ready to sign a complaint against them (or until they had cheated enough
to have it proved against them).<br />
<h2>
<span class="mw-headline" id="My_response">My response</span></h2>
The final is entirely multiple choice. I got the results files from
the scantron office. I figured that I wouldn't quite know what to do
with a comparison just between these two kids (unless the tests were
identical), and that it would be just about as easy (and far more
informative) to compare <i>everybody</i> to <i>everybody else</i>. It's
still kind of hard for me to get used to the fact that we have computers
now and can really do stuff like this. I calculated the number of
identical right answers and the number of identical wrong answers for
each pair of students (~18K pairs), and plotted it out.<br />
<span class="ww-dynamic-project-file ww-dynamic-project-file-processed"><span class="ww-altlinks noprint"><span class="ww-altlinks-pulldown-arrow"></span></span><img alt="(cplot.Rout-0.png)" src="http://lalashan.mcmaster.ca/theobio/math/index.php/Special:GetProjectFile?project=Answer_matching&make=false&display=raw&random-number=367&filename=cplot.Rout-0.png" /></span><br />
The line corresponds to forty total shared answers (two students
having identical test papers). This did not happen. But there were four
points near the line that looked like clear outliers to me:<br />
<span class="ww-dynamic-project-file ww-dynamic-project-file-processed"><span class="ww-altlinks noprint"><span class="ww-altlinks-pulldown-arrow"></span></span><img alt="(cplot.Rout-1.png)" src="http://lalashan.mcmaster.ca/theobio/math/index.php/Special:GetProjectFile?project=Answer_matching&make=false&display=raw&random-number=316&filename=cplot.Rout-1.png" /></span><br />
<h2>
<span class="mw-headline" id="The_follow_up">The follow up</span></h2>
I wasn't sure what to do next, but the registrar's office knew. They make <i>seating maps</i> during exams. They didn't offer to help out, but I was allowed to go and examine the maps.<br />
The results were amazing.<br />
<ul>
<li><i>All four</i> of the identified pairs were seated adjacent (three
pairs were side by side, and the fourth pair had one student behind the
other). The probability that this might have happened by chance is
beyond ridiculous.</li>
</ul>
<ul>
<li><i>None</i> of the four identified pairs were seated in the room
where the alert invigilators hassled the pair of cheaters. This might
have been by chance, but I doubt it. Likely the invigilators in the
other room were visibly less alert.</li>
</ul>
I talked to the academic integrity office, and various experts, and
figured out that it really was impossible to be sure who had cheated in
the side-by-side pairs. I did put all 6 of them through a bit of an
ordeal, though, and at least half of them deserved it. I was also unable
to convict the person in front of the front-back pair (although it's
hard to see how that one would have worked without collusion). The
person in the back of the front-back pair denied all knowledge, but
received a zero for the exam grade plus a confidential, temporary
notation of my finding at the integrity office (the strongest punishment
I was allowed to give). They promised to fight it, but never did.<br />
<h2>
<span class="mw-headline" id="Postscript">Postscript</span></h2>
I now use versioning, but I'm starting to discover that this does not
necessarily prevent cheating, either. I may have more adventures to
report, soon.<br />
I definitely get the feeling that the person I caught cheated their
way through Mac. The initial response to my call was pretty relaxed.
They did get an F in my class (I couldn't give an automatic F for the
class, but the exam zero was sufficient). They retook the class and
passed, expunging the F, and graduating presumably with a clean record.<br />
I have heard a lot of anecdotal reports of people dealing with
cheating informally (or not at all). It's kind of depressing. My
impression is that Mac has a cheating problem, and we need to fight
back.<br />
<h2>
<span class="mw-headline" id="Code">Code</span></h2>
<br />
The code used to produce these plots in R is shown <a href="http://lalashan.mcmaster.ca/theobio/math/index.php/Answer_matching/comparisons" title="Answer matching/comparisons">here</a>.Jonathan Dushoffhttp://www.blogger.com/profile/03197609136976213552noreply@blogger.com62tag:blogger.com,1999:blog-5234436246664688381.post-41945618097113286912014-12-08T09:08:00.001-08:002014-12-08T09:08:20.574-08:00Third Aunt<div style="background-color: white; font-family: sans-serif; font-size: 12.6666669845581px; line-height: 19.0499992370605px; margin-bottom: 0.5em; margin-top: 0.4em;">
If I tell you that Corinthia is my second cousin, once removed, there are two major problems:</div>
<ul style="background-color: white; font-family: sans-serif; font-size: 12.6666669845581px; line-height: 19.0499992370605px; list-style-image: url(http://lalashan.mcmaster.ca/theobio/math/skins/stealthninja/bullet.gif?2014-11-06T14:58:20Z); list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;">You probably don't know what the heck that means</li>
</ul>
<ul style="background-color: white; font-family: sans-serif; font-size: 12.6666669845581px; line-height: 19.0499992370605px; list-style-image: url(http://lalashan.mcmaster.ca/theobio/math/skins/stealthninja/bullet.gif?2014-11-06T14:58:20Z); list-style-type: square; margin: 0.3em 0px 0px 1.6em; padding: 0px;">
<li style="margin-bottom: 0.1em;">If you do know, you still don't know whether she is one generation older, or one generation younger.</li>
</ul>
<div style="background-color: white; font-family: sans-serif; font-size: 12.6666669845581px; line-height: 19.0499992370605px; margin-bottom: 0.5em; margin-top: 0.4em;">
This nomenclature has been around for a long time, and it's just not working. So we need a new system. From now on, we'll all use the system outlined below.</div>
<div style="background-color: white; font-family: sans-serif; font-size: 12.6666669845581px; line-height: 19.0499992370605px; margin-bottom: 0.5em; margin-top: 0.4em;">
<br /></div>
<div style="background-color: white; font-family: sans-serif; font-size: 12.6666669845581px; line-height: 19.0499992370605px; margin-bottom: 0.5em; margin-top: 0.4em;">
<a href="http://lalashan.mcmaster.ca/theobio/math/index.php/Third_aunt">See more</a></div>
Jonathan Dushoffhttp://www.blogger.com/profile/03197609136976213552noreply@blogger.com0tag:blogger.com,1999:blog-5234436246664688381.post-9342805422927981852014-12-04T13:25:00.005-08:002015-02-20T10:22:36.498-08:00Factoring integers using the complex plane<i>Please see <a href="http://lalashan.mcmaster.ca/theobio/math/index.php/Complex_factoring"> my wiki</a> for a comprehensible version of this post.</i><br />
<br />
I was pretty pleased that I factored a recent <a class="external text" href="http://lalashan.mcmaster.ca/theobio/math/index.php/Composite_of_the_day" rel="nofollow">Composite of the day</a> entirely in the complex plane.<br />
The number was 9509, which I noticed immediately is 97²+10². Since I know <a class="external text" href="http://en.wikipedia.org/wiki/Fermat%27s_little_theorem" rel="nofollow">Fermat's little theorem</a>,
and I know that the Composite of the day is composite, I knew there
should be another way to write it as the sum of two squares. A little
bit of counting (100+193+191) showed that it is also equal to 95²+22².<br />
<br />
For some reason, I know that if I use those two summations to write
complex integers with modulus 9509, their greatest common factor will
also divide 9509.<br />
So I said, (97+10i) - (95+22i) = 2-12i. The modulus of that is
2²+12²=148. The factors of 2 must be irrelevant (since the Cotd is odd),
so <b>37 </b>should be the number we're looking for.<br />
<br />
Similarly, (97-10i) - (95+22i) = 2-32i. The modulus of that is 1048.
Again, discarding factors of 2, we're left with the prime <b>257</b>.<br />
<br />
And the number is now factored, by finding complex
integers with the right modulus and manuipulating them in the complex
plane.<br />
<br />
<br />
Pretty wild, huh?Jonathan Dushoffhttp://www.blogger.com/profile/03197609136976213552noreply@blogger.com2tag:blogger.com,1999:blog-5234436246664688381.post-34099211762269408352014-07-22T06:48:00.002-07:002014-07-22T06:48:39.532-07:00Geostationary orbitsI was thinking about gravity and space elevators today. They still
confuse me. So I decided to see if I could work out the height of a
geostationary orbit in my head, while walking to school. This despite
the fact that I don't remember any of the classical mechanics stuff I
learned in college, but with the powerful ally of ... dimensional
analysis, which is approximately the coolest thing ever. I got it badly
wrong, and later figured out why.<br />
<br />
<h2>
<span class="mw-headline" id="What_we_know">What we know</span></h2>
<ul>
<li><a class="ww-tex-math-link" href="http://lalashan.mcmaster.ca/theobio/math/index.php/Special:GetProjectFile?project=Standalone%3FGeostationary_orbits%3Fww-internal-1a9002ec85642a643bb7ea48ed7403a5.tex-math&make=false&filename=ww-internal-1a9002ec85642a643bb7ea48ed7403a5.tex-math.latexml.html5.make.log"><span class="latexml"><span class="MathJax_Preview"></span><span aria-label="g=10\mathrm{m}/\mathrm{s}^{2}" aria-readonly="true" class="MathJax" id="MathJax-Element-1-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-1" style="display: inline-block; width: 5.176em;"><span style="display: inline-block; font-size: 103%; height: 0px; position: relative; width: 5.006em;"><span style="clip: rect(1.669em, 1000em, 3.208em, -0.673em); left: 0em; position: absolute; top: -2.731em;"><span class="mrow" id="MathJax-Span-2"><span class="mrow" id="MathJax-Span-3"><span class="mi" id="MathJax-Span-4" style="font-family: MathJax_Math; font-style: italic;">g<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-5" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mrow" id="MathJax-Span-6" style="padding-left: 0.278em;"><span class="mrow" id="MathJax-Span-7"><span class="mn" id="MathJax-Span-8" style="font-family: MathJax_Main;">10</span><span class="mo" id="MathJax-Span-9"></span><span class="mi" id="MathJax-Span-10" style="font-family: MathJax_Main;">m</span></span><span class="mo" id="MathJax-Span-11" style="font-family: MathJax_Main;">/</span><span class="msup" id="MathJax-Span-12"><span style="display: inline-block; height: 0px; position: relative; width: 0.833em;"><span style="clip: rect(2.055em, 1000em, 2.969em, -0.65em); left: 0em; position: absolute; top: -2.731em;"><span class="mi" id="MathJax-Span-13" style="font-family: MathJax_Main;">s</span><span style="display: inline-block; height: 2.731em; width: 0px;"></span></span><span style="left: 0.379em; position: absolute; top: -2.714em;"><span class="mn" id="MathJax-Span-14" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.351em; width: 0px;"></span></span></span></span></span></span></span><span style="display: inline-block; height: 2.731em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.273em; overflow: hidden; vertical-align: -0.336em; width: 0px;"></span></span></nobr></span></span></a></li>
<li><a class="ww-tex-math-link" href="http://lalashan.mcmaster.ca/theobio/math/index.php/Special:GetProjectFile?project=Standalone%3FGeostationary_orbits%3Fww-internal-2f5e5db0b7f6c71c64c7002c023a1371.tex-math&make=false&filename=ww-internal-2f5e5db0b7f6c71c64c7002c023a1371.tex-math.latexml.html5.make.log"><span class="latexml"><span class="MathJax_Preview"></span><span aria-label="r=6.4\mathrm{Mm}" aria-readonly="true" class="MathJax" id="MathJax-Element-2-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-15" style="display: inline-block; width: 5.176em;"><span style="display: inline-block; font-size: 103%; height: 0px; position: relative; width: 5.006em;"><span style="clip: rect(1.82em, 1000em, 2.98em, -0.662em); left: 0em; position: absolute; top: -2.731em;"><span class="mrow" id="MathJax-Span-16"><span class="mrow" id="MathJax-Span-17"><span class="mi" id="MathJax-Span-18" style="font-family: MathJax_Math; font-style: italic;">r</span><span class="mo" id="MathJax-Span-19" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mrow" id="MathJax-Span-20" style="padding-left: 0.278em;"><span class="mn" id="MathJax-Span-21" style="font-family: MathJax_Main;">6.4</span><span class="mo" id="MathJax-Span-22"></span><span class="mi" id="MathJax-Span-23" style="font-family: MathJax_Main; padding-left: 0.167em;">Mm</span></span></span></span><span style="display: inline-block; height: 2.731em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.882em; overflow: hidden; vertical-align: -0.101em; width: 0px;"></span></span></nobr></span></span></a></li>
<li><a class="ww-tex-math-link" href="http://lalashan.mcmaster.ca/theobio/math/index.php/Special:GetProjectFile?project=Standalone%3FGeostationary_orbits%3Fww-internal-de56b86221775b626077153215d2b4fe.tex-math&make=false&filename=ww-internal-de56b86221775b626077153215d2b4fe.tex-math.latexml.html5.make.log"><span class="latexml"><span class="MathJax_Preview"></span><span aria-label="C=24\mathrm{h}/2\pi=14\mathrm{ks}" aria-readonly="true" class="MathJax" id="MathJax-Element-3-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-24" style="display: inline-block; width: 8.77em;"><span style="display: inline-block; font-size: 103%; height: 0px; position: relative; width: 8.495em;"><span style="clip: rect(1.753em, 1000em, 3.208em, -0.633em); left: 0em; position: absolute; top: -2.731em;"><span class="mrow" id="MathJax-Span-25"><span class="mrow" id="MathJax-Span-26"><span class="mi" id="MathJax-Span-27" style="font-family: MathJax_Math; font-style: italic;">C<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.045em;"></span></span><span class="mo" id="MathJax-Span-28" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mrow" id="MathJax-Span-29" style="padding-left: 0.278em;"><span class="mrow" id="MathJax-Span-30"><span class="mrow" id="MathJax-Span-31"><span class="mn" id="MathJax-Span-32" style="font-family: MathJax_Main;">24</span><span class="mo" id="MathJax-Span-33"></span><span class="mi" id="MathJax-Span-34" style="font-family: MathJax_Main;">h</span></span><span class="mo" id="MathJax-Span-35" style="font-family: MathJax_Main;">/</span><span class="mn" id="MathJax-Span-36" style="font-family: MathJax_Main;">2</span></span><span class="mo" id="MathJax-Span-37"></span><span class="mi" id="MathJax-Span-38" style="font-family: MathJax_Math; font-style: italic;">π<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.003em;"></span></span></span><span class="mo" id="MathJax-Span-39" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mrow" id="MathJax-Span-40" style="padding-left: 0.278em;"><span class="mn" id="MathJax-Span-41" style="font-family: MathJax_Main;">14</span><span class="mo" id="MathJax-Span-42"></span><span class="mi" id="MathJax-Span-43" style="font-family: MathJax_Main;">k</span><span class="mo" id="MathJax-Span-44"></span><span class="mi" id="MathJax-Span-45" style="font-family: MathJax_Main;">s</span></span></span></span><span style="display: inline-block; height: 2.731em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.186em; overflow: hidden; vertical-align: -0.336em; width: 0px;"></span></span></nobr></span></span></a></li>
</ul>
I chose to start with 10 and 6.4 (instead of, say, 9.8 and 6) because
I thought I would want to take the square root of their product.<br />
<br />
<br />
<a class="ww-tex-math-link" href="http://lalashan.mcmaster.ca/theobio/math/index.php/Special:GetProjectFile?project=Standalone%3FGeostationary_orbits%3Fww-internal-0d61f8370cad1d412f80b84d143e1257.tex-math&make=false&filename=ww-internal-0d61f8370cad1d412f80b84d143e1257.tex-math.latexml.html5.make.log"><span class="latexml"><span class="MathJax_Preview"></span><span aria-label="C" aria-readonly="true" class="MathJax" id="MathJax-Element-4-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-46" style="display: inline-block; width: 0.801em;"><span style="display: inline-block; font-size: 103%; height: 0px; position: relative; width: 0.758em;"><span style="clip: rect(1.646em, 1000em, 2.828em, -0.633em); left: 0em; position: absolute; top: -2.579em;"><span class="mrow" id="MathJax-Span-47"><span class="mi" id="MathJax-Span-48" style="font-family: MathJax_Math; font-style: italic;">C<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.045em;"></span></span></span><span style="display: inline-block; height: 2.579em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.905em; overflow: hidden; vertical-align: -0.101em; width: 0px;"></span></span></nobr></span></span></a>
is the characteristic time of the Earth's rotation (how long it takes a
point on the surface to travel the length of the radius). In my
experience, the characteristic time (not the period) tends to be the
quantity that gives you the right answer in dimensional analysis.<br />
<br />
<h2>
<span class="mw-headline" id="The_simple_answer">The simple answer</span></h2>
We have too much information for a good dimensional analysis (too
many ways of combining our quantities to get the right units). But there
does seem to be a natural, straightforward way to do it.<br />
<a class="ww-tex-math-link" href="http://lalashan.mcmaster.ca/theobio/math/index.php/Special:GetProjectFile?project=Standalone%3FGeostationary_orbits%3Fww-internal-60c8a45b19c328a159f585a4934d77f5.tex-math&make=false&filename=ww-internal-60c8a45b19c328a159f585a4934d77f5.tex-math.latexml.html5.make.log"><span class="latexml"><span class="MathJax_Preview"></span><span aria-label="\sqrt{gr}=8\mathrm{km}/\mathrm{s}" aria-readonly="true" class="MathJax" id="MathJax-Element-5-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-49" style="display: inline-block; width: 6.113em;"><span style="display: inline-block; font-size: 103%; height: 0px; position: relative; width: 5.916em;"><span style="clip: rect(1.753em, 1000em, 3.351em, -0.611em); left: 0em; position: absolute; top: -2.731em;"><span class="mrow" id="MathJax-Span-50"><span class="mrow" id="MathJax-Span-51"><span class="msqrt" id="MathJax-Span-52"><span style="display: inline-block; height: 0px; position: relative; width: 1.854em;"><span style="clip: rect(1.909em, 1000em, 3.011em, -0.673em); left: 0.834em; position: absolute; top: -2.579em;"><span class="mrow" id="MathJax-Span-53"><span class="mrow" id="MathJax-Span-54"><span class="mi" id="MathJax-Span-55" style="font-family: MathJax_Math; font-style: italic;">g<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-56"></span><span class="mi" id="MathJax-Span-57" style="font-family: MathJax_Math; font-style: italic;">r</span></span></span><span style="display: inline-block; height: 2.579em; width: 0px;"></span></span><span style="clip: rect(3.522em, 1000em, 4.118em, -0.683em); left: 0.834em; position: absolute; top: -4.357em;"><span style="display: inline-block; height: 0px; position: relative; width: 1.019em;"><span style="font-family: MathJax_Main; left: -0.084em; position: absolute; top: -4.02em;">−<span style="display: inline-block; height: 4.02em; width: 0px;"></span></span><span style="font-family: MathJax_Main; left: 0.325em; position: absolute; top: -4.02em;">−<span style="display: inline-block; height: 4.02em; width: 0px;"></span></span></span><span style="display: inline-block; height: 4.02em; width: 0px;"></span></span><span style="clip: rect(2.992em, 1000em, 4.448em, -0.611em); left: 0em; position: absolute; top: -3.827em;"><span style="font-family: MathJax_Main;">√</span><span style="display: inline-block; height: 4.02em; width: 0px;"></span></span></span></span><span class="mo" id="MathJax-Span-58" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mrow" id="MathJax-Span-59" style="padding-left: 0.278em;"><span class="mrow" id="MathJax-Span-60"><span class="mn" id="MathJax-Span-61" style="font-family: MathJax_Main;">8</span><span class="mo" id="MathJax-Span-62"></span><span class="mi" id="MathJax-Span-63" style="font-family: MathJax_Main;">k</span><span class="mo" id="MathJax-Span-64"></span><span class="mi" id="MathJax-Span-65" style="font-family: MathJax_Main;">m</span></span><span class="mo" id="MathJax-Span-66" style="font-family: MathJax_Main;">/</span><span class="mi" id="MathJax-Span-67" style="font-family: MathJax_Main;">s</span></span></span></span><span style="display: inline-block; height: 2.731em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.333em; overflow: hidden; vertical-align: -0.483em; width: 0px;"></span></span></nobr></span></span></a>
is a speed. This should have something to do with something. Multiply
by 14ks to get 112Mm. This seems way too high, though, so I should think
this through more carefully.<br />
<br />
I wonder if <a class="ww-tex-math-link" href="http://lalashan.mcmaster.ca/theobio/math/index.php/Special:GetProjectFile?project=Standalone%3FGeostationary_orbits%3Fww-internal-8715566d8eb6be677a48e14c00a5cdf2.tex-math&make=false&filename=ww-internal-8715566d8eb6be677a48e14c00a5cdf2.tex-math.latexml.html5.make.log"><span class="latexml"><span class="MathJax_Preview"></span><span aria-label="8\mathrm{km}/\mathrm{s}" aria-readonly="true" class="MathJax" id="MathJax-Element-6-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-68" style="display: inline-block; width: 2.832em;"><span style="display: inline-block; font-size: 103%; height: 0px; position: relative; width: 2.731em;"><span style="clip: rect(1.753em, 1000em, 3.208em, -0.64em); left: 0em; position: absolute; top: -2.731em;"><span class="mrow" id="MathJax-Span-69"><span class="mrow" id="MathJax-Span-70"><span class="mrow" id="MathJax-Span-71"><span class="mn" id="MathJax-Span-72" style="font-family: MathJax_Main;">8</span><span class="mo" id="MathJax-Span-73"></span><span class="mi" id="MathJax-Span-74" style="font-family: MathJax_Main;">k</span><span class="mo" id="MathJax-Span-75"></span><span class="mi" id="MathJax-Span-76" style="font-family: MathJax_Main;">m</span></span><span class="mo" id="MathJax-Span-77" style="font-family: MathJax_Main;">/</span><span class="mi" id="MathJax-Span-78" style="font-family: MathJax_Main;">s</span></span></span><span style="display: inline-block; height: 2.731em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.186em; overflow: hidden; vertical-align: -0.336em; width: 0px;"></span></span></nobr></span></span></a> is orbital velocity or escape velocity and the dimensional analysis discovered that by accident.<br />
<br />
<h2>
<span class="mw-headline" id="The_right_answer">The right answer</span></h2>
If we're moving away from the surface of the Earth, we have to respect our knowledge that gravity goes as <a class="ww-tex-math-link" href="http://lalashan.mcmaster.ca/theobio/math/index.php/Special:GetProjectFile?project=Standalone%3FGeostationary_orbits%3Fww-internal-294ce890c4a3b9529d80d402420ecd46.tex-math&make=false&filename=ww-internal-294ce890c4a3b9529d80d402420ecd46.tex-math.latexml.html5.make.log"><span class="latexml"><span class="MathJax_Preview"></span><span aria-label="r^{2}" aria-readonly="true" class="MathJax" id="MathJax-Element-7-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-79" style="display: inline-block; width: 0.957em;"><span style="display: inline-block; font-size: 103%; height: 0px; position: relative; width: 0.91em;"><span style="clip: rect(1.138em, 1000em, 2.438em, -0.662em); left: 0em; position: absolute; top: -2.2em;"><span class="mrow" id="MathJax-Span-80"><span class="msup" id="MathJax-Span-81"><span style="display: inline-block; height: 0px; position: relative; width: 0.909em;"><span style="clip: rect(1.909em, 1000em, 2.817em, -0.662em); left: 0em; position: absolute; top: -2.579em;"><span class="mi" id="MathJax-Span-82" style="font-family: MathJax_Math; font-style: italic;">r</span><span style="display: inline-block; height: 2.579em; width: 0px;"></span></span><span style="left: 0.455em; position: absolute; top: -2.714em;"><span class="mn" id="MathJax-Span-83" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.351em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.2em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.026em; overflow: hidden; vertical-align: -0.089em; width: 0px;"></span></span></nobr></span></span></a> to make use of <a class="ww-tex-math-link" href="http://lalashan.mcmaster.ca/theobio/math/index.php/Special:GetProjectFile?project=Standalone%3FGeostationary_orbits%3Fww-internal-b2f5ff47436671b6e533d8dc3614845d.tex-math&make=false&filename=ww-internal-b2f5ff47436671b6e533d8dc3614845d.tex-math.latexml.html5.make.log"><span class="latexml"><span class="MathJax_Preview"></span><span aria-label="g" aria-readonly="true" class="MathJax" id="MathJax-Element-8-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-84" style="display: inline-block; width: 0.488em;"><span style="display: inline-block; font-size: 103%; height: 0px; position: relative; width: 0.455em;"><span style="clip: rect(1.909em, 1000em, 3.011em, -0.673em); left: 0em; position: absolute; top: -2.579em;"><span class="mrow" id="MathJax-Span-85"><span class="mi" id="MathJax-Span-86" style="font-family: MathJax_Math; font-style: italic;">g<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.003em;"></span></span></span><span style="display: inline-block; height: 2.579em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.823em; overflow: hidden; vertical-align: -0.289em; width: 0px;"></span></span></nobr></span></span></a>, so we need to construct <a class="ww-tex-math-link" href="http://lalashan.mcmaster.ca/theobio/math/index.php/Special:GetProjectFile?project=Standalone%3FGeostationary_orbits%3Fww-internal-4c26e4d9651c3e7325936780b5a4af61.tex-math&make=false&filename=ww-internal-4c26e4d9651c3e7325936780b5a4af61.tex-math.latexml.html5.make.log"><span class="latexml"><span class="MathJax_Preview"></span><span aria-label="K=gr^{2}" aria-readonly="true" class="MathJax" id="MathJax-Element-9-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-87" style="display: inline-block; width: 3.77em;"><span style="display: inline-block; font-size: 103%; height: 0px; position: relative; width: 3.641em;"><span style="clip: rect(1.669em, 1000em, 3.163em, -0.652em); left: 0em; position: absolute; top: -2.731em;"><span class="mrow" id="MathJax-Span-88"><span class="mrow" id="MathJax-Span-89"><span class="mi" id="MathJax-Span-90" style="font-family: MathJax_Math; font-style: italic;">K<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.04em;"></span></span><span class="mo" id="MathJax-Span-91" style="font-family: MathJax_Main; padding-left: 0.278em;">=</span><span class="mrow" id="MathJax-Span-92" style="padding-left: 0.278em;"><span class="mi" id="MathJax-Span-93" style="font-family: MathJax_Math; font-style: italic;">g<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-94"></span><span class="msup" id="MathJax-Span-95"><span style="display: inline-block; height: 0px; position: relative; width: 0.909em;"><span style="clip: rect(1.909em, 1000em, 2.817em, -0.662em); left: 0em; position: absolute; top: -2.579em;"><span class="mi" id="MathJax-Span-96" style="font-family: MathJax_Math; font-style: italic;">r</span><span style="display: inline-block; height: 2.579em; width: 0px;"></span></span><span style="left: 0.455em; position: absolute; top: -2.714em;"><span class="mn" id="MathJax-Span-97" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.351em; width: 0px;"></span></span></span></span></span></span></span><span style="display: inline-block; height: 2.731em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.226em; overflow: hidden; vertical-align: -0.289em; width: 0px;"></span></span></nobr></span></span></a>. Translating the prefixes back to km gives us one extra 1000, so we have <a class="ww-tex-math-link" href="http://lalashan.mcmaster.ca/theobio/math/index.php/Special:GetProjectFile?project=Standalone%3FGeostationary_orbits%3Fww-internal-166c08553f6fa5b0e3913f886676dfbb.tex-math&make=false&filename=ww-internal-166c08553f6fa5b0e3913f886676dfbb.tex-math.latexml.html5.make.log"><span class="latexml"><span class="MathJax_Preview"></span><span aria-label="400,000\mathrm{km}^{3}/\mathrm{s}^{2}" aria-readonly="true" class="MathJax" id="MathJax-Element-10-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-98" style="display: inline-block; width: 6.816em;"><span style="display: inline-block; font-size: 103%; height: 0px; position: relative; width: 6.599em;"><span style="clip: rect(1.669em, 1000em, 3.208em, -0.655em); left: 0em; position: absolute; top: -2.731em;"><span class="mrow" id="MathJax-Span-99"><span class="mrow" id="MathJax-Span-100"><span class="mn" id="MathJax-Span-101" style="font-family: MathJax_Main;">400</span><span class="mo" id="MathJax-Span-102" style="font-family: MathJax_Main;">,</span><span class="mrow" id="MathJax-Span-103" style="padding-left: 0.167em;"><span class="mrow" id="MathJax-Span-104"><span class="mn" id="MathJax-Span-105" style="font-family: MathJax_Main;">000</span><span class="mo" id="MathJax-Span-106"></span><span class="mi" id="MathJax-Span-107" style="font-family: MathJax_Main;">k</span><span class="mo" id="MathJax-Span-108"></span><span class="msup" id="MathJax-Span-109"><span style="display: inline-block; height: 0px; position: relative; width: 1.289em;"><span style="clip: rect(2.061em, 1000em, 2.958em, -0.658em); left: 0em; position: absolute; top: -2.731em;"><span class="mi" id="MathJax-Span-110" style="font-family: MathJax_Main;">m</span><span style="display: inline-block; height: 2.731em; width: 0px;"></span></span><span style="left: 0.834em; position: absolute; top: -2.714em;"><span class="mn" id="MathJax-Span-111" style="font-family: MathJax_Main; font-size: 70.7%;">3</span><span style="display: inline-block; height: 2.351em; width: 0px;"></span></span></span></span></span><span class="mo" id="MathJax-Span-112" style="font-family: MathJax_Main;">/</span><span class="msup" id="MathJax-Span-113"><span style="display: inline-block; height: 0px; position: relative; width: 0.833em;"><span style="clip: rect(2.055em, 1000em, 2.969em, -0.65em); left: 0em; position: absolute; top: -2.731em;"><span class="mi" id="MathJax-Span-114" style="font-family: MathJax_Main;">s</span><span style="display: inline-block; height: 2.731em; width: 0px;"></span></span><span style="left: 0.379em; position: absolute; top: -2.714em;"><span class="mn" id="MathJax-Span-115" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.351em; width: 0px;"></span></span></span></span></span></span></span><span style="display: inline-block; height: 2.731em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 1.273em; overflow: hidden; vertical-align: -0.336em; width: 0px;"></span></span></nobr></span></span></a>.<br />
The radius of the earth doesn't really directly affect the orbit. In
fact, I used it only because I know it, and I don't know the mass of the
Earth or the gravitational constant <a class="ww-tex-math-link" href="http://lalashan.mcmaster.ca/theobio/math/index.php/Special:GetProjectFile?project=Standalone%3FGeostationary_orbits%3Fww-internal-dfcf28d0734569a6a693bc8194de62bf.tex-math&make=false&filename=ww-internal-dfcf28d0734569a6a693bc8194de62bf.tex-math.latexml.html5.make.log"><span class="latexml"><span class="MathJax_Preview"></span><span aria-label="G" aria-readonly="true" class="MathJax" id="MathJax-Element-11-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-116" style="display: inline-block; width: 0.801em;"><span style="display: inline-block; font-size: 103%; height: 0px; position: relative; width: 0.758em;"><span style="clip: rect(1.646em, 1000em, 2.828em, -0.633em); left: 0em; position: absolute; top: -2.579em;"><span class="mrow" id="MathJax-Span-117"><span class="mi" id="MathJax-Span-118" style="font-family: MathJax_Math; font-style: italic;">G</span></span><span style="display: inline-block; height: 2.579em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.905em; overflow: hidden; vertical-align: -0.101em; width: 0px;"></span></span></nobr></span></span></a>. This means that the right answer must be made from <a class="ww-tex-math-link" href="http://lalashan.mcmaster.ca/theobio/math/index.php/Special:GetProjectFile?project=Standalone%3FGeostationary_orbits%3Fww-internal-a5f3c6a11b03839d46af9fb43c97c188.tex-math&make=false&filename=ww-internal-a5f3c6a11b03839d46af9fb43c97c188.tex-math.latexml.html5.make.log"><span class="latexml"><span class="MathJax_Preview"></span><span aria-label="K" aria-readonly="true" class="MathJax" id="MathJax-Element-12-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-119" style="display: inline-block; width: 0.957em;"><span style="display: inline-block; font-size: 103%; height: 0px; position: relative; width: 0.91em;"><span style="clip: rect(1.668em, 1000em, 2.806em, -0.652em); left: 0em; position: absolute; top: -2.579em;"><span class="mrow" id="MathJax-Span-120"><span class="mi" id="MathJax-Span-121" style="font-family: MathJax_Math; font-style: italic;">K<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.04em;"></span></span></span><span style="display: inline-block; height: 2.579em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.86em; overflow: hidden; vertical-align: -0.078em; width: 0px;"></span></span></nobr></span></span></a> and <a class="ww-tex-math-link" href="http://lalashan.mcmaster.ca/theobio/math/index.php/Special:GetProjectFile?project=Standalone%3FGeostationary_orbits%3Fww-internal-0d61f8370cad1d412f80b84d143e1257.tex-math&make=false&filename=ww-internal-0d61f8370cad1d412f80b84d143e1257.tex-math.latexml.html5.make.log"><span class="latexml"><span class="MathJax_Preview"></span><span aria-label="C" aria-readonly="true" class="MathJax" id="MathJax-Element-13-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-122" style="display: inline-block; width: 0.801em;"><span style="display: inline-block; font-size: 103%; height: 0px; position: relative; width: 0.758em;"><span style="clip: rect(1.646em, 1000em, 2.828em, -0.633em); left: 0em; position: absolute; top: -2.579em;"><span class="mrow" id="MathJax-Span-123"><span class="mi" id="MathJax-Span-124" style="font-family: MathJax_Math; font-style: italic;">C<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.045em;"></span></span></span><span style="display: inline-block; height: 2.579em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 0.905em; overflow: hidden; vertical-align: -0.101em; width: 0px;"></span></span></nobr></span></span></a>, which means in turn that there's only one way to do it: <a class="ww-tex-math-link" href="http://lalashan.mcmaster.ca/theobio/math/index.php/Special:GetProjectFile?project=Standalone%3FGeostationary_orbits%3Fww-internal-57290e34b6069c853643be5c89012ae0.tex-math&make=false&filename=ww-internal-57290e34b6069c853643be5c89012ae0.tex-math.latexml.html5.make.log"><span class="latexml"><span class="MathJax_Preview"></span><span aria-label="(KC^{2})^{{1/3}}" aria-readonly="true" class="MathJax" id="MathJax-Element-14-Frame" role="textbox"><nobr><span class="math" id="MathJax-Span-125" style="display: inline-block; width: 4.629em;"><span style="display: inline-block; font-size: 103%; height: 0px; position: relative; width: 4.475em;"><span style="clip: rect(0.565em, 1000em, 3.077em, -0.503em); left: 0em; position: absolute; top: -2.2em;"><span class="mrow" id="MathJax-Span-126"><span class="msup" id="MathJax-Span-127"><span style="display: inline-block; height: 0px; position: relative; width: 4.474em;"><span style="clip: rect(2.415em, 1000em, 4.67em, -0.503em); left: 0em; position: absolute; top: -3.792em;"><span class="mrow" id="MathJax-Span-128"><span class="mo" id="MathJax-Span-129" style="vertical-align: 0em;"><span style="font-family: MathJax_Size2;">(</span></span><span class="mrow" id="MathJax-Span-130"><span class="mi" id="MathJax-Span-131" style="font-family: MathJax_Math; font-style: italic;">K<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.04em;"></span></span><span class="mo" id="MathJax-Span-132"></span><span class="msup" id="MathJax-Span-133"><span style="display: inline-block; height: 0px; position: relative; width: 1.276em;"><span style="clip: rect(1.646em, 1000em, 2.828em, -0.633em); left: 0em; position: absolute; top: -2.579em;"><span class="mi" id="MathJax-Span-134" style="font-family: MathJax_Math; font-style: italic;">C<span style="display: inline-block; height: 1px; overflow: hidden; width: 0.045em;"></span></span><span style="display: inline-block; height: 2.579em; width: 0px;"></span></span><span style="left: 0.822em; position: absolute; top: -2.783em;"><span class="mn" id="MathJax-Span-135" style="font-family: MathJax_Main; font-size: 70.7%;">2</span><span style="display: inline-block; height: 2.351em; width: 0px;"></span></span></span></span></span><span class="mo" id="MathJax-Span-136" style="vertical-align: 0em;"><span style="font-family: MathJax_Size2;">)</span></span></span><span style="display: inline-block; height: 3.792em; width: 0px;"></span></span><span style="left: 3.337em; position: absolute; top: -3.228em;"><span class="mrow" id="MathJax-Span-137"><span class="mn" id="MathJax-Span-138" style="font-family: MathJax_Main; font-size: 70.7%;">1</span><span class="mo" id="MathJax-Span-139" style="font-family: MathJax_Main; font-size: 70.7%;">/</span><span class="mn" id="MathJax-Span-140" style="font-family: MathJax_Main; font-size: 70.7%;">3</span></span><span style="display: inline-block; height: 2.351em; width: 0px;"></span></span></span></span></span><span style="display: inline-block; height: 2.2em; width: 0px;"></span></span></span><span style="border-left: 0em solid; display: inline-block; height: 2.274em; overflow: hidden; vertical-align: -0.747em; width: 0px;"></span></span></nobr></span></span></a>.<br />
<br />
This is a pain to calculate mentally.<br />
<br />
The computer claims it's 43Mm, which still seems way too high, so what's up?<br />
<br />
Checking <a class="external text" href="http://en.wikipedia.org/wiki/Geosynchronous_orbit" rel="nofollow">wikipedia</a>, it turns out that's what's <i>up</i>
is the geostationary orbit, which really is 42Mm above the center of
the Earth (or 36Mm above your head). Which is wild, but score one at
least for dimensional analysis.<br />
<br />
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<div class="mw-content-ltr" dir="ltr" id="mw-content-text" lang="en">
<span class="ww-dynamic-project-file ww-dynamic-project-file-processed"><span class="ww-altlinks noprint ww-write-only"><span class="ww-altlinks-pulldown-arrow"></span></span></span><br />
<br />
This post was developed on <a class="external text" href="http://lalashan.mcmaster.ca/theobio/projects/index.php/WorkingWiki" rel="nofollow">WorkingWiki</a> at <a href="http://lalashan.mcmaster.ca/theobio/math/index.php/Geostationary_orbits" title="Geostationary orbits">Geostationary orbits</a>. The version there may be newer (or have better links).</div>
Jonathan Dushoffhttp://www.blogger.com/profile/03197609136976213552noreply@blogger.com0tag:blogger.com,1999:blog-5234436246664688381.post-28013011266970988272014-07-14T20:35:00.002-07:002014-07-15T07:22:46.047-07:00Soccer mania!I obviously need to work on my blogging skills.<br />
<div>
<br /></div>
<div>
It looks like nobody figured out my Sucker Bet question, but I think a lot of people glanced at it (and the early comments), and <i>thought</i> that they had. Of course, it may be that somebody figured it out after I posted the answer, because after all what would you say if that happened? Nonetheless, it's worth another glance, IMHO.</div>
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<br /></div>
<div>
In honor of the World Cup, I'm posting this "self-generating puzzle". How many Group B results can be worked out from the information on <a href="http://www.uefa.com/womenseuro/season=2013/standings/index.html">this page</a> alone:? It's a lot of fun to read the sports pages and find phrases or tables that work as puzzles by themselves, although I rarely have time anymore.</div>
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<br /></div>
<div>
Here's another self-generating sports puzzle. Making reasonable assumptions about how sportswriters write, how many results can be inferred from the following sentence?</div>
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<span style="font-family: Courier New, Courier, monospace;">After last night's win, the Broncos have won 2 of their last 3 games, and 5 of their last 7.</span></div>
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<span style="font-family: Courier New, Courier, monospace;"><br /></span></div>
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<span style="font-family: inherit;">I don't think I am the one who discovered this puzzle, but I was unable to figure out who did by searching usenet archives.</span></div>
Jonathan Dushoffhttp://www.blogger.com/profile/03197609136976213552noreply@blogger.com2tag:blogger.com,1999:blog-5234436246664688381.post-70262881760562582012014-07-08T02:12:00.000-07:002019-03-14T07:20:45.275-07:00Sucker bet<a class="external text" href="http://www.mawode.com/blog/" rel="nofollow">Walt</a> sent me this puzzle (reworded from <a class="external text" href="http://cacm.acm.org/magazines/2013/2/160169-puzzled-tumbling-dice/fulltext" rel="nofollow">this set of great puzzles from Communications of the ACM</a>; you can get it by accessing from a University or library).<br />
<br />
Alice and Bob roll 2 standard 6-sided dice, note their sum, and repeat.
Alice wins if a 7 is rolled, and then followed immediately by another 7.
Bob wins if an <i>8</i> is followed immediately by a 7. They continue rolling until somebody wins. Who has the better odds of winning?<br />
<br />
Of course the answer is the non-intuitive one. Can you figure out why? As a person with a long-time fondness for craps (I know lots of people who are no good at probability, <i>except</i> when it involves two dice), this seems to me like the ultimate sucker bet to offer someone.<br />
<br />
<span class="ww-dynamic-project-file ww-dynamic-project-file-processed">More discussion (and a link to three ways to think about the answer) on <a href="http://dushoff.github.io/notebook/sucker.html">JD's notebook.</a></span>Jonathan Dushoffhttp://www.blogger.com/profile/03197609136976213552noreply@blogger.com9