What Price A Stamp?

So stamp prices are going up again. And by "going" I mean "went, like 5 months ago". The wheels of Project Potpourri grind slowly, but they grind exceedingly fine.

Now they have a Forever Stamp. You buy it today and it will be good until the end of time regardless of future price increases. Sounds like a great way to save literally hundreds of cents, right? My wife (or maybe it was me--these human details are so much dross) posed this question: If I buy a 41 cent Forever Stamp in 2007, am I really saving money vs a 53 cent stamp in 2020 or does it come out in the inflationary wash?

IIRC, inflation is something like 3% per year, but of course we don't have the information on what future stamp prices are going to be. What about history? How closely have stamp prices tracked inflation in the past? Using a history of stamp prices and relative dollar values since 1913 I was able to answer this question. (Notes: All this is based on the first ounce only. I didn't try to go back farther than 1913. For each year, I only use the average dollar value for the year, I didn't break it down by month. For years with more than one price increase, I only report the last one. Since 2007 isn't over yet, I used the last dollar value available, which is September.)

Year	Apparent price that year	Stamp price in 2007 dollars	% diff relative to today
1917	0.03	0.49	16
1919	0.02	0.24	-70
1932	0.03	0.46	10
1958	0.04	0.29	-42
1963	0.05	0.34	-20
1968	0.06	0.36	-14
1971	0.08	0.41	0
1974	0.10	0.42	3
1975	0.13	0.50	18
1978	0.15	0.48	14
1981	0.20	0.46	10
1985	0.22	0.43	3
1988	0.25	0.44	6
1991	0.29	0.44	7
1995	0.32	0.44	6
1999	0.33	0.41	0
2001	0.34	0.40	-2
2002	0.37	0.43	4
2006	0.39	0.40	-1
2007	0.41	0.41	0

It used to fluctuate quite a bit, but for almost the last 30 years, they've been within a few percents of the same price. And given that even a 10% difference is only $.04, I'd call it fairly constant over the last 100 years (except for a few years).

How To: Lose Weight (And I Don't Even Mention Lettuce!)

First of all, let's define our terms. There's "losing weight" and there's "getting healthy". For the latter, you need to eat carrots and exercise. I'm not too interested in carrots and, while I don't mind incidental exercise, I don't have the time or inclination to run around for no direct reason. This post is solely about making the number on the scale be smaller. That in itself is a great step towards "getting healthy", though, as long as you aren't too stupid about it (i.e. no starvation).

The mantra in diet books is "don't diet--change your lifestyle". Partly this is just a good idea. They don't want you going on a crash diet and then fattening back up. But partly this is making a virtue of necessity. The reason they want you to change your lifestyle is that a non-starvation diet doesn't change your weight fast enough to notice it unless you try it over the long term. That is, if you only drop .4 lbs in a week, are you really going to notice it adding up even over the course of a month, if you last that long? Who is going to remember, to the tenth of a pound, what they weighed 2 weeks ago? The first secret of losing weight: You need to keep a history of your progress to refer back to.

But if you are only losing .4 lbs/week, there's another problem: Noise. The key to controlling a variable to to be able to measure it accurately. Otherwise how do you know if what you are trying is working? But diet books also tell you not to weigh yourself very often. The reason they give for this is that your weight can vary because of non-fat variables. A large meal only partially digested, extra water, etc.

That's really terrible advice, though. For instance, children's test scores vary a lot from child to child, so should you just choose one random child from each school to measure performance? Of course not! If anything, taking fewer measurements increases the noise problem. The way to fix noisy data is to remove the noise. The second secret of losing weight: Noise reduction.

One simple way to remove noise from data is via averaging. Particularly, a "moving average". Let's say I take the following daily measurements:

Mon: 201
Tue: 201
Wed: 202
Thu: 201
Fri: 201
Sat: 202
Sun: 200
Mon: 201
Tue: 200
Wed: 200
Thu: 199
Fri: 200
Sat: 201

Nothing is happening! This diet sucks!!!

But wait, let's try doing a moving average. For each day, we'll average in with the previous two days (which means we have to skip the first two since there aren't two days before them).

Mon: 201 (no avg)
Tue: 201 (no avg)
Wed: 202 (201.3)
Thu: 201 (201.3)
Fri: 201 (201.3)
Sat: 202 (201.3)
Sun: 200 (201)
Mon: 201 (201)
Tue: 200 (200.3)
Wed: 200 (200.3)
Thu: 199 (199.6)
Fri: 200 (199.6)
Sat: 201 (200)

In fact, I lost a pound to 1.3 pounds, depending on where you count from. (There are many ways to doing a moving average, including ways to weight the average more heavily towards more recent measurements. Don't worry about the specific method here.)

The difference between raw and averaged (aka "smoothed") data is even more dramatic if you look at a graph. The circles are (fictitious-but-realistic) readings from the scale. The line is the smoothed average. Some of those weigh-ins differ by as much as 2 pounds in a single day. If you wake up a day after "being good" on your diet and see you weigh 2 pounds more than yesterday, doesn't that make you want to give up? But after you smooth the data, the problem may not be so bad.

In fact, it might not be a problem at all. Say the 3 days you were averaging yesterday were 205,202,201 (202.6). All you ate yesterday was carrots, but today you got 203. The weighted average is still 202, which is down from yesterday's average. How awesome is that!

Answer: Very awesome. But not so awesome we need to make things harder for ourselves. For instance, try to weigh yourself under the same circumstances every time: Same time of day, same state of undress, same fullness of stomach and bladder, etc. Also, I have gotten into the habit of "unofficially" weighing myself at various times throughout the day and I've gotten to know exactly how much to subtract for my clothing, how much water I'll lose via respiration overnight, etc. If I weigh X when I got to bed, I will weigh between X-3.5 and X-3 in the morning, rock-solid. An unofficial weigh-in the evening before can help prevent sticker shock in the morning and will also tell you if you can afford a bowl of ice cream. (That might not work for you.)

But who wants to juggle a bunch of numbers?! That's worse than being fat! Don't worry, you don't have to do a thing. Just head on over to PhysicsDiet. Create a free account, give it some info like your starting weight (you can ignore all the stuff about percent bodyfat and exercise) and away you go. The site handles the moving average and plots pretty charts and everything.

Since I started in late March I've only been losing an average of .44 lbs/wk. That's slow enough that I would have given up after a couple weeks, especially since it's also far less than the noisiness of the data (particularly since my scale only weighs in .5 lb increments). But with the two secrets of losing weight, historical progress and noise reduction, I've managed to lose over 15 lbs so far. Also, there are 3500 calories to a pound of fat, so that's just an average of 220 calories per day, which I barely even notice missing from my plate, let alone do I have to eat carrots and rice cakes. It's like I'm not even dieting (almost).

Christmas Present Idea

Long have I pined for a mechanical watch where you could actually see the gears and springs and whatnot flying around. I've seen things like this with the "Fossil" brand attached, but they are always less than what I really wanted. I tried googling for "mechanical watch with visible gears" but that wasn't too helpful.

Today I discovered the magic word: Skeleton. A "skeleton" watch is one where it is purposely made to be see-through. "Full skeleton" means that even things that you'd kind of like to have on a watch, like numbers and hands, are as tiny and hard to use as possible so you can see the full glory of the gears.

Feast your image display software on THIS:

I gather it is possible to spend up to and including ONE MILLION DOLLARS on a watch like this, but, I further gather, Chinese imports can be had for under $100. Even well under $100. For instance.

I wouldn't have many requirements on such a gift. It should be as visible as possible and be actually mechanical, no hidden battery doing the work.

A New Recipe for π

I've been working on a Difference Engine in Lego. A little derivative perhaps, but still a big challenge. For one thing, there's not that much construction detail at that site. For another, what little there is I'm ignoring. I want to try to solve this on my own.

I've made some progress, but my (borrowed) video camera is being cranky so I've been unable to record and post it. (Aside to person I borrowed it from: I'm just getting a black screen in record mode. Also a little red flashing light that I think is the button battery so I thought that was it. However recording suddenly started work despite that, but only for a few minutes. ???) Thus this post isn't about that.

When the Difference Engine actually is running, I thought it would be fun to have it calculate π. NO WAIT, LET ME FINISH!!! I know π is transcendental, meaning there is no polynomial for which π is the solution.

The point of the Difference Engine is that as you crank the handle, you calculate the value of the polynomial for higher and higher values of x. What I'd like is a polynomial such that for higher and higher x, the value is closer and closer to π.

So I just google for a polynomial that does that, right? I mean, there must be hundreds of them by now. No. There are none that I can find.

There are plenty of series approximations, however. For instance:

π = ¹/₁ - ¹/₃ + ¹/₅ - ¹/₇ + ¹/₉ - ¹/₁₁ +....

And with some series...es, it's possible to come up with an expression that the series sums to up to any given point. For instance, the sum of the first n odd numbers:

1 + 3 + 5 + 7 + .... 2n-1 = n²

So maybe one of the series approximations of π can be manipulated into a polynomial. Then I can use the Lego Difference Engine on that polynomial and crank out 2 or 3 digits.

However, I'm having a great deal of trouble with this (not that that indicates anything other than the fact that I'm really not that great at math). For one thing, I've concentrated my efforts on the simple-to-remember 1/1 - 1/3 + 1/5 approach. But I just realized this morning that this series alternately overshoots and undershoots the target, meaning it has an infinite number of humps and valleys. A polynomial of degree n can have a maximum of n-1 humps and valleys, AFAIK, so that series is out.

If I'm going to start from a series, I need one that's always more or less than π and never the other. Or an entirely different idea. I can think of plenty of iterative methods, but that's basically just a series. I need a single step where the accuracy is chosen by the value of x I input. Since I haven't been able to find any reference to such a thing, I'm thinking it hasn't ever been done. Is that because it's impossible?

You Program My Back, I'll Program Yours

If you are a math/science/computer nerd and you have a child, you have undoubtedly wondered how you can teach your child programming. I googled for such a thing more than once and found the usual suspects: BASIC, LOGO, etc, etc. The free ones were all half-finished or too hard, the good ones were all expensive or geared towards classrooms.

I tried teaching Number One Son (8 years old) some pseudo-codey stuff to do simple math problems and learn about loops. He enjoyed that, but we didn't get very far and I always had to be the virtual machine to check if his program ran.

However, MIT has recently come up with something that absolutely rules--Scratch. And it's free!

Scratch is graphical. You drag the little components around to assemble a program. For instance, to make a loop, you drag your components into a loop widget, which wraps around it like a vice. If you want to construct a conditional, you get out the "if" widget and drag and drop logical/mathematical conditions in from the toolbox. Just fill in the blanks and go.

The GUI isn't just for show, either. You don't feel like you are using the mouse to write a program, you feel like you are literally assembling a physical object. And it eliminates syntax errors, which is a major deal in the under-13 crowd. Furthermore, the graphical programming language ties right in to the very graphics-oriented programs Scratch is targeted towards and children love. Creating and animating sprites takes just a few clicks. Object collision is just a matter of checking if two colors are touching. And this is all clearly presented enough that an 8 year old can (and has) figured most of it out himself.

I think he learned more about programming in 2 days with Scratch than he did in all the previous years of my bumbling explanations. He goes off and works on a program for a while and then will come to me with a question about how to do something. And they are pretty sophisticated problems (considering his age), such as how to cycle through sprite costumes and wraparound at the end or how to keep various sprites in sync. With the concrete example of his non-working program providing the motivation, the explanations of modular arithmetic or semaphores stick much better.

So far he's created programs that simulate a robot in a maze, animate a rocket flying to the moon, teach the alphabet to his two-year-old sibling and even one generic drawing program with adjustable pen size and color. All 100% on his own.

Windows and Mac only, but:

1. They claim they'll have a Linux version out "before the end of 2007".
2. The usefulness, fun and polish of Scratch is more than worth setting up an old PC with Windows.

It's possible to upload your program to share with other kids, but we haven't tried that yet. Videos of Scratch in action..

Lego Marble Pump II: Handcranked Bugaloo

The basic idea behind the Lego marble pump is working. I cobbled it together well enough to demonstrate My Vision, but many kinks remain to be worked out. For instance, the little levers need to be worked automatically. But I'm not going to do that, I'm just going to post the video of it hobbling along.

In case that isn't clear:

1. Marble goes in ramp at left
2. Cam on central carriage hits lever to open gate, allowing marble into carriage (part of cam played by my finger)
3. Central carriage lifts, eventually dumping marble into next higher ramp (another "cam" lifts lever there to make sure gate is closed)
4. Central carriage lowers, goto 2

Difference Engineering

I've known about Babbage's Difference and Analytical Engines for a long time, but never known much about how they work. Finally I actually read a little information on the first one (in the context of Legos) and it's pretty interesting from both a mathematical and mechanical point of view. Based on the information on that page, I worked out a tiny additional step of my own (though surely Babbage himself already knew this).

First of all, the basic idea: The purpose of the Difference Engine was to pre-calculate tables of polynomials for books in the days before portable calculators. So you'd want to known 3x³ + 14x² + 5 for all values of x from 1 to, say, 1000. As it happens, there's a clever little shortcut such that if you have a few values you can calculate the next one very simply using only addition from the previous answer (thus "Difference Engine").

So let's say f(x) = x² + 3x.

x   f(x)   diff1  diff2 
1     4      -      -   
2    10      6      -   
3    18      8      2
4    28     10      2
5    40     12      2

See that second difference column is a constant. If we'd picked a polynomial of the 3rd degree, we'd have to work this out to the 3rd column. Fourth degree, 4th column, etc.

So to build a machine, all you need to do is set the value for x = 1 and set the Nth difference in the last column. It automatically adds that difference to the previous difference, which is cascaded upwards until f(x + 1) is arrived at. Then you go around again. All that is required is simple addition.

What isn't explained on that page (that I saw) was how to know ahead of time what that last difference is going to be. Sure, you can work out N rows to get that Nth column, but it would be nice if you could set the inputs from direct inspection of the polynomial. As a matter of fact, this is easy.

Say f(x) = ax² + bx + c. Then the difference between two successive answers (i.e. the value in the first difference column) is going to be:

a(x+1)² + b(x+1) + c - ax² - bx - c
= ax² + 2ax + a + bx + b + c - ax² - bx - c
= 2ax + a + b

The difference between successive entries in THAT column are going to be:

2a(x+1) + a + b - 2ax - a - b
= 2ax + 2a + a + b - 2ax - a - b
= 2a

And if we look at our example, we did indeed get 2 * 1 as the constant in the last column.

Working this out for a 3rd degree polynomial gives 6a (where a is the coefficient of x³). Based on these two examples and looking at Pascal's Triangle, I predicted the value for a 4th degree would be 20. But it was actually 24.

In fact, if you work it out it should be clear that the constant difference will be a * n!, where a is the coefficient of the highest power of x and n is that power. It should be simple to prove this using induction, since all other terms always drop out and you muliply the coefficient by the power at each step on the way down.

So if I wanted to know what the constant difference was in the 5th column of differences for the polynomial 8x⁵ - 3x³ + 117, I just multiple 8 times 5! and get 960. I cram that, plus the initial value for x = 1 onto the machine and get cranking.

Lego Marble Pump

This is a pretty neat machine. The core of the pump mechanism is pretty genius and I built one myself based on that graphic. The problem is that the tower can only be as high as the number of marbles you have and the number of marbles your motor can lift.

The most common way of lifting balls in a contraption is with a kind of one-way elevator. But I've already built one of those. This guy has an interesting idea. It's a little hard to see in the video, but he's got the marble going both ways through the pistons. It lifts one "story" going from left to right, then another story from right to left.

That gave me an idea. First of all, why not any number of stories? And second of all, why three pistons? Probably the answer to both is: maximum height for the amount of Lego he had.

I'm thinking you could put just one piston between two towers and with a certain amount of mechanical futzing pump the marbles up to any height.

*pause for 24 hours of trying to get some kind of Lego CAD software working on any computer in my house or office to illustrate this and failing miserably*

OK, here's the basic idea: The piston in the middle has ramps inside. The marble starts off at the bottom of one of the side towers. It rolls into the piston down one of the ramps and it stopped against the wall of the other tower. The piston moves up and there's a little nook in the other tower that the marble falls into. The piston lowers. Now there's another internal ramp, this time pointed the other direction, aligned with the nook, so the marble re-enters the piston and is stopped against the wall of the first tower. Repeat.

It's a little tricky, though. The nook is the problem. The marble needs to roll out of a ramp into the nook but then roll out of the nook into a ramp. The three piston video solved this by having good timing and long ramps. I don't have long ramps or the patience to get the timing right. What I'm trying right now is a kind of teeter-totter arrangement in the nook that tips up to receive the marble and then tips down to release it.

Oh cripes. I just re-read this entry and even I was too bored by the last two paragraphs to pay attention to them. I'll just have to make it work and post a video. Or somehow get LDraw working.

The Temperature of What?

So I had all these posts about temperature logging and one post with the actual logged temperatures...but the temperature of what?

A solar hot box!

You probably already know what this is, but just in case you don't: It's basically a tiny greenhouse. Or like a car left in the sun at noon in August. Only it's even hotter, since it is insulated, painted black and pointed right at the sun.

Inside the box I put a jar with 250 ml of cooking oil and poked a hole in the lid for a temperature probe. That's what these temps are.

Why did I choose cooking oil? Because I didn't want evaporation to be a problem. For one thing, it would fog up the inside of the glass. For another, it would cap my max temperature at 100°C (not that that turned out to be a problem in this case). And lastly, it would change the amount of water in the bottle and I needed that to be a constant because I did some calculations with it.

Knowing the amount of oil and the temperature change (plus looking up the specific heat of vegetable oil), I can calculate the rate at which energy is entering the oil. For the above graph, I got 2.5 watts for the steepest part of the curve. However, I see that the site I just linked to has the specific heat of veg oil as 1.67 kJ/kg K and I was using 2.5. So maybe the power is really more like 3.7 watts.

Knowing the area of the collector I can also calculate the amount of power falling into the box. That's about 75 watts. So the end-to-end efficiency was only about 3-5%. Not that great.

Imagine if you put a cup of water on the table and then turn the furnace thermostat up to 90°. How much energy are you going to waste before the water gets hot? This illustrates the 3 main problems:

1. Air passively surrounding a container of liquid isn't going to heat it very fast.
2. There's a lot of volume of air being heated uselessly.
3. During all this time, heat is escaping the cracks, windows, chimney, etc. In the case of the hot box, the glass front gets very hot and is radiating a lot of the energy right back out.

If a hot box is like an oven, the next version will be like a microwave. Don't heat up the air, just beam energy right into the substance.

Locomotive Lego Linkage

It seems like I keep having to reinvent this linkage. The basic idea is that I want to turn a crank (i.e. rotary motion) which makes something move back and forth (i.e. linear motion). You see this exact same linkage on classic train wheels, though in that case the linear motion is the primary mover and it is turning into rotary motion to move the train.

In fact, almost all piston engines do this same thing. What I keep not realizing is that you need to hold the straight part straight or it doesn't work. Not "it doesn't go straight" but not work at all.

Let me show you what I mean

I think the reason I keep forgetting this is that the movement looks so smooth. It doesn't seem like there could be that much friction there, but there must be. The piston is rubbing against the sides. Not (just) because of a tight fit, but out of sheer necessity to make the piston run straight.

There are linkages that have been invented that can convert rotary motion to linear without all the side friction. For instance, here. A 3D version of that linkage is in the Boston Museum of Science. Another picture of Peaucellier is here and the page also calls my "locomotive linkage" a "sewing machine".

Peaucellier isn't very practical because it requires so many parts, which take up space and can fail. Or can it be made simply? Can another practical, mathematically accurate, low-friction linkage be built? In Lego?