Mathematical table turning

This paper at the front for the Mathematics ArXiv describes a fascinating, unintuitive result: under certain (fairly lax) conditions, a four-legged rectangular table can always be rotated around its center so that all four legs are on the surface, that is, the table does not wobble. From the paper:

We prove that if the ground does not rise by more than arctan(1/sqrt(2)) ≈ 35.26° between any two of its points, and if the legs of the table are at least half as long as its diagonals, then the table can be balanced anywhere on the ground, without any part of it digging into the ground, by turning the table on the spot.

The paper also discusses leveling tables in the real world. The result does not hold for tables with different leg lenths, and tables whose leg endpoints are not strictly rectangular. Still, it is a curious finding.

HP49G+ Advanced User's Reference Manual

Gene Wright announced a new version of the 49G+ manual, and it is an impressive piece of work, reminiscent in quality and scope of the HP manuals from the golden age. Contributors include Gene Wright, Tony Hutchins, Wlodek Mier-Jedrzejowicz, Jordi Hidalgo, Ted Kerber, Joe Horn, Richard Nelson and Jake Schwartz.

TI Basic Coding on the PC, Made Easier

You can use TI's GraphLink software to edit TI Basic code on the PC, but you can't use GraphLink to transfer the code to the calculator through a USB connection, because GraphLink doesn't support the USB cable. Here's a workaround.

Open GraphLink, edit your code and save it. Open a Windows file explorer window where you saved the code file. Right-click on the filename and choose 'Send to TI Device ...', then send the file as usual. You can leave both the explorer window and GraphLink open, continue to edit your code & send it to the calculator. This method assumes that you have TI Connect installed, so the 'Send to TI Device ...' option is available.

Ideally TI will eventually update TI Connect to include a program editor. The old GraphLink program made it very easy to develop code, because you could download the code directly from the editor.

Also related to TI Connect and USB, I often have difficulty getting a connection between PC and calculator. I believe this is caused by a flakey USB driver or motherboard implementation, rather than a fault in TI Connect, but I seem to have better luck when I plug the calculator in first, then the USB cable at the PC.

MAA November Columns

Keith Devlin addresses more fundamental probability confusions. A detailed-enough article about pushing the Shannon limit of error free communications. These MAA columns are usually easy to understand and interesting.

Colored bubbles and disappearing dye

This Popular Science article describes Tim Kehoe's eleven-year saga to develop colored soap bubbles.

Peanut Butter

My older daughter really likes peanut butter, which makes this essay by W.F. Buckley relevant.

slashdot in 100 words or less

From a /. poster:

If the sample of Slashdot and its immediate social clique were the norm, we'd live in a pseudosocialist utopia in which all of us are gainfully employed and paid a hundred thousand dollars to work 30 hour weeks developing beautiful open source software that we give away and nobody buys, and all music and entertainment is produced through the honest labor of talented people upon whom we benevolently bestow voluntary payments for their work, and whose labors of love are distributed for free through the software channels that we were paid lots of money to develop. Oh, and Bush isn't president. And global warming stopped. And we all ride bikes to our jobs. And there's no McDonald's or suburbs. And soda is free. So is beer. I could go on, but I moved into the TrollZone about 5 minutes ago.

Panexa

The drug for everyone. A little too close to the truth.

Umberto Eco

Nice site here. From the essay The Future of the Book:

Books will remain indispensable not only for literature, but for any circumstance in which one needs to read carefully, not only to receive information but also to speculate and to reflect about it.

Cheap tool alert - 4in Digital Calipers

American Science and Surplus' latest catalog has a 4" digital caliper for about $15US. I could not resist the temptation, as I've been looking for a small digital caliper to keep in my small toolbox in my office at work.

The caliper is made in China, according to the box, but there is no manufacturer's name. The construction is completely plastic and feels a bit flimsy, but should hold up with care. The plastic construction is an advantage in that the caliper is very light, which is nice if your toolbox is already on the heavy side. The moving blade head has noticable wobble, but less than the resolution of 0.01". The moving head slides smoothly on the beam. Overall length is 7". The outside measurement blade depth is about 1-3/16", and the inside measurement blade depth is 1/2". Unlike most calipers, there is no depth measurement pin. There is also no locking mechanism.

With a display digit height of 3/8" and good contrast, measurements are easy to read. The display is a little slow to update while the head is in motion, but not objectionably. Three color-coded buttons control the OFF, ON and ZERO functions. The ON button is also used to toggle between inch and millimeter measurement. There is no tactile click to the buttons but they are easy to operate, regardless.

The resolution is only 0.01" and 0.1mm which is less than the more typical resolution of 0.001". I checked the accuracy against my Mitutoyo CD-6CS calipers, and measurements are consistently low across the range by 0.01", which isn't bad. The indicated metric measurement matches the inch measurement to the displayed resolution. Using the metric indication gives you more than twice the resolution (0.01" vs. 0.004"), but the error on the metric scale is still about 0.01" low.

The caliper is powered by an SR44 watch battery, which is included. The caliper will automatically power-down after a few minutes of inactivity. Remember to re-zero the caliper after changing the battery, because it seems to power up with a random offset. In fact, it is a good idea to check the zero frequently, because I noticed an occasional offset of about 0.2" had occurred.

If you don't already have a caliper, I can't really recommend this one for general purpose use. The limited range, low resolution, lack of a depth pin and occasional random offsets all mean that your money would be better put towards a better 6" caliper. However, if, like me, you just need a spare to keep in a satellite toolbox, this one will do the job. It will also be less heartbreaking if someone walks off with this caliper than your $150 'good one'. This caliper would certainly be better than nothing if you are on a tight tool budget.

At this price, this would be a good caliper for hacking applications, if you need to build a displacement measurement display into a fixture or instrument. I haven't had it apart, but it might be possible to tap into the circuit to transmit the measurement to another instrument.

A thoroughly good AMS Notices this month

The December Notices of the AMS is up here. We have nasty high-dimensional integrals made somewhat less nasty, random matrices, a mathematical institute in the Palo Alto Fry's, a sensible proposal to eliminate college-level math homework, Martin Gardner arguing for realism, and E.B. Davies arguing against it.

Free registration required; documents are PDF.

Forman S. Acton's 'Real Computing Made Real'

Forman S. Acton's Real Computing Made Real is now available from Dover, in paperback, at a reasonable price. This book is in some respects a follow-up to, and an amplication of the author's previous 'Numerical Methods That (usually) Work', but it stands on its own quite well. If you liked 'Numerical Methods', you'll like 'Real Computing' and learn some new things.

'Real Computing' ostensibly focuses on solving nonlinear equations and performing numerical integration, but the real value is in teaching how to recoginze difficult problems, and how to reformulate the problem to get a good answer. The pace is leisurely and thorough; not at all dry but rather engaging. This seems to offend readers who prefer a more pedantic presentation. Here's a sample:


AN EDUCATIONAL DALLIANCE

If you've never spent a numerical night with a robust alternating series, here's your chance! Compute sin(x) from

sin(x) = x - x^3/3! + x^5/5! - ...

Evaluate each term from its predecessor, summing as you go. Stop when you term shrinks to insignificance. Check your technique with x = 0.5 -- then go for x = 9.5 and x = 11. The experience seldom thrills but is very maturing -- especially if you are using a calculator with less than 10-digit precision.


To a calculator user of canned solvers and integrators, this book still offers much. It will teach you to recognize equations and integrals which are difficult to numerically solve quickly, accurately, or both. You will also learn some general methods to recast those nasty problems to get better results. 'Real Computing' is not an introduction to numerical methods nor is it a cookbook. Dover does in fact offer several good introductions:

A First Course in Numerical Analysis, Ralston & Rabinowitz
Numerical Methods, Dahlquist and Bjork
Introduction to Numerical Analysis, Hildbrand
Numerical Methods for Scientists and Engineers, Hamming


Acton devotes about thirty pages to sketching functions by hand. The motivation is to learn to really understand how the terms of a function affect its form, and to use that understanding to apply an appropriate algorithm or tranformation. However, much of that understanding could be acquired using the built-in function plotting features of graphing calculators.

(I have no business connection with Dover; just a happy repeat customer.)

Range limitation in TIbasic rand() function

rand(n) returns a random integer in the range [1,n]. The maximum (undocumented) value for 'n' is 99,999,999,999,999, or only 14 digits. Would be handy to have a function to extend the argument range to the full integer range of 640 digits or so. The function should retain the good statistical properties of the built-in rand() function, so the question becomes: what is the best way to combine multiple results from rand() to build a larger integer? Probably already plenty of results for this already; need to search. Also remember to handle negative arguments for 'n'.

193-digit RSA-640 Factored

Here's the MathWorld write-up. The v200 (eventually) correctly returns false from isPrime() with this number. Didn't try factoring it, though ...

Popularity of US first names

This interactive plot of US first names is interesting and well-done, from a graphics standpoint. You can see how the popularity of various names has waxed and waned over the years. I came across it from this MAA article. Needs Java.

The coin change problem

What are the best denominations of 4 coin values to minimize the coins needed to make any value from one cent to 99 cents? How optimum is are the US values of 25, 10, 5 and 1 cent? Are there better values? How much does adding the 50 cent coin improve the situation?

Wrote a TIbasic function misc\change(val_list,amt) which returns the number of coins of each denomination needed to make 'amt'. val_list is a list of descending coin denominations. The penalty function P is the sum of the number of coins to make each amount from 1 cent to 100 cents. For example, misc\change({25,10,5,1},99) returns {3, 2, 0, 4}, for a total of 9 coins. Some coin values & P values:

{25,10,5,1} 470
{50,25,10,5,1} 420 (50cent piece improves P by 50)
{32,8,4,1} 450
{32,16,4,1} 450

pcpress() comments

Playing with pcpress() file compression on v200. Some problems:

- key bounce; more than one key event per keypress
- tiny font is hard to read
- failed to write variable for large (~40K) archived variable.