tag:artscience.cyberclip.com,2013:/posts Art+Science 2017-02-18T17:42:22Z Paul Clip tag:artscience.cyberclip.com,2013:Post/341928 2012-03-05T01:14:38Z 2013-10-08T16:34:44Z Learning Glassblowing
I recently took a series of three intro to glassblowing courses at Revere Glass in Berkeley, California. Our teachers were very helpful, the workshop itself well stocked with tools, and the glass was plentiful. The fish sculpture below is the output of my third course.

Glassblowing has proved to be an interesting combination of art and science. We'd seen its master craftsmen at work in Venice and shaping glowing, flowing, 1000C hot glass proved as challenging and fun as I'd imagined. You can let your creativity loose yet still have to respect and be conscious of the physical properties of glass. The difference between cold and hot seals, flashing your piece to prevent cracking, constantly rotating the glass to maintain control over it, how to work "frit" or colors into your piece, and much more.

All in all it was a great experience and, if you're in the Bay Area, I highly recommend Revere Glass (watch out for their Groupon offers).

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Paul Clip
tag:artscience.cyberclip.com,2013:Post/341932 2011-10-23T22:03:53Z 2013-10-08T16:34:44Z Mount Diablo Tarantulas
As I was biking up Mount Diablo the other day, I came across an animal I'd been keeping an eye out for: a male tarantula in wanderlust. They're easiest to spot as they cross the road, and this one was quite determined to get to the other side.

In the fall the male tarantulas set off in search of females to mate with. If she doesn't eat him (which she'll only do if she's famished) he'll keep looking for partners until the cold weather, or a hungry lover, gets him. Females, on the other hand, can live up to twenty years.

I've always liked spiders but this one was wary of me and wouldn't stay on my hand. That's OK :-)

This article has more information about these fascinating creatures.

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Paul Clip
tag:artscience.cyberclip.com,2013:Post/341933 2011-06-08T07:41:58Z 2017-02-18T17:42:22Z World Population by Time Zone Cory Doctorow, scifi author and BoingBoing co-founder, once wrote a scifi novel called Eastern Standard Tribe (available free). It was fun read but what I enjoyed most was his idea that people would belong to a "tribe" based on their time zone. In Doctorow's world, your loyalties lie not with the country of your birth but with the people who are up when you are.

This evening I was thinking about this novel and idly wondered which tribe would be the biggest? In other words, which time zone is the most highly populated?

Looking at a map, the answer was obvious: it had to be UTC+8 which includes not only China but Malaysia, the Philippines, and more.

Still, the fun of such a frivolous question is less in the answer than in the answering, so I fired up Mathematica and a few calculations later generated this graph.

I cheated a little by using a simplifying assumption: if a country has multiple time zones, I divide its population evenly between them. This inaccuracy doesn't change the fact that our top three are... <drumroll>
  1. UTC+8: China and others
  2. UTC+5.5: India and others
  3. UTC+1: Western Europe and a good chunk of Africa

According to Mathematica, there are 39 different time zones ranging from UTC-11.5 to UTC+14. I wonder if anyone has visited them all? Now that would be a glorious adventure! :-)
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Paul Clip
tag:artscience.cyberclip.com,2013:Post/341937 2011-05-31T05:02:00Z 2013-10-08T16:34:44Z Fearless Fire Eating at The Crucible

My 12 year old son Thomas and I took a three hour fire eating course yesterday in downtown Oakland at The Crucible. We found it through the excellent Workshop Weekend program, which offers many interesting courses. We were the only two students and our teacher, Patricia, gave us a great introduction to both the art and science of fire eating

Warning: fire eating is dangerous and an easy way to get hurt quickly. The information below is no substitute for proper instruction (it's incomplete too!). In other words... Don't do this at home!

After a comprehensive review of safety precautions, Patricia taught us how to make our own torches. These consisted of 18" aluminum rods with a wick at one end. Interestingly, the wick is made of a 12" long strip of kevlar and held in place by kevlar string. Kevlar has high heat resistance and is reasonably absorbent so it makes for a great wick. 

For our initial foray into fire eating, we used rubbing alcohol as fuel because the flame is small. Maybe so but we were still more than a little apprehensive about putting a flaming torch in our mouths!

After the first try it became a lot easier and we soon graduated to white (camping) gas which generates much bigger and brighter flames, as the pictures below show. We quickly conquered any fears we had and became pretty comfortable eating fire.

The science behind this art wasn't what I'd expected. I'd assumed that fire eating consisted of closing your mouth around the flaming torch to deprive it of oxygen and so stop it burning. Not so. You never fully close your mouth around the torch (burnt lips anyone?) instead you close them partially and exhale to extinguish the flame.

Once our basic skills were in place we moved on to art. Patricia taught us various tricks such as flame transfers and ways to light the torches. She was particularly impressed by Thomas who, in the six years she's been teaching the class, was by far her youngest student.

All in all it was indeed a glorious adventure and one we're going to practice ourselves, safely.

 

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Paul Clip
tag:artscience.cyberclip.com,2013:Post/341942 2011-03-29T05:45:00Z 2013-10-08T16:34:44Z Stunning Norwegian Auroras

Love the way this was filmed. The smooth panning transforms the time lapse pictures. Norway is such a beautiful country (if a little cold ;-)

Auroras are the result of the solar wind colliding with gases in the earth's upper atmosphere (more details). A wonderful combination of art and science.

Source
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Paul Clip
tag:artscience.cyberclip.com,2013:Post/341958 2011-03-16T23:56:00Z 2016-04-17T01:46:35Z Visualizing One Hundred Years of Pacific Rim Earthquakes
Whenever I hear of a major earthquake, I always wonder when our turn will come. I've been asking myself that question way too frequently recently. My family and I live in the San Francisco Bay Area, prime earthquake country (or so I thought until I looked at Japan...). I started playing with NOAA's earthquake data after the New Zealand earthquake. After the recent Japan quake, I thought I'd publish a few graphics. 

Disclaimer: I'm no geologist, statistician, or expert on earthquakes. I don't even play one on TV. I don't think anyone can predict earthquakes with any certainty (though there is some interesting research) and I certainly won't try.

The Ring of Fire is the name given to the chain of mountains, volcanoes, and faults that ring the Pacific Ocean. Of the world's 16 largest earthquakes since 1900, 15 occurred in the Ring of Fire.

Here's an interesting graphic showing the earthquakes above 6.0 magnitude that have hit the Ring of Fire region since 1900. Earthquakes of magnitudes between 6 and 7 are in green, between 7 and 8 in blue, and 8 or higher in red.

Notice anything? Well, as a Californian, the first thing that struck me was: "we're getting off lightly!".

This movie gives you a different way to see the earthquakes. Same legend as before: magnitude 6+ green, 7+ blue, 8+ red. The video isn't the most exciting one you'll ever see. It helps to pick a point of interest on the map and imagine some elevator music in the background :-)

Let's dig deeper...

Japan, California, New Zealand, and Chile

Looking at the graph above it's clear California gets fewer earthquakes than many other parts of the Ring.

When I compare a circular area 2,000km around the center of Japan with the same size area around California (centered on San Francisco), Japan has been hit four times as often by large (i.e. 6+ magnitude) earthquakes than California (~200 vs. ~50).

Here are Japan's large earthquakes, with 6.x, 7.x, and 8+ magnitude earthquakes broken out (notice that the data for 6.x earthquakes in 1900-1950 is likely incomplete):

The equivalent map for the 2,000 km surrounding San Francisco looks like this (sorry, no, there are no 8+ earthquakes, NOAA has the 1906 one at 7.9):

Let's look at the "earthquake history" in the other recent hotspots: New Zealand and Chile.

Earthquakes from 1900-2011 in a 2,000km area centered on Christchurch, New Zealand.

And finally South America. 2,000km area centered on Santiago, Chile. (That 9.5'er in 1960 was a monster). 

BTW, I've only focused on a few of the Ring of Fire hotspots. Indonesia, Central America, etc. are all very active.

So are we Californians due for an earthquake?

As I wrote earlier: Who really knows? On the one hand 110 years of data tells us that our corner of the Ring of Fire experiences 25% as many earthquakes as Japan. On the other hand... It may be about time for a big one to hit us.

Simon Winchester (an author whose many books I'd recommend, esp. The Man Who Loved Chinawrote recently:

[The Chile, New Zealand, and Japan earthquakes]  involved more or less the same family of circum-Pacific fault lines and plate boundaries—and though there is still no hard scientific evidence to explain why, there is little doubt now that earthquakes do tend to occur in clusters: a significant event on one side of a major tectonic plate is often—not invariably, but often enough to be noticeable—followed some weeks or months later by another on the plate’s far side. [...]

Now there have been catastrophic events at three corners of the Pacific Plate—one in the northwest, on Friday; one in the southwest, last month; one in the southeast, last year.

That leaves just one corner unaffected—the northeast.

Are earthquakes really clustered? I haven't analyzed the data for correlations. Just eyeballing the graphs above, there are enough earthquakes happening around the Pacific Rim that you could claim some correlation exists.

If you want my advice... Better safe than sorry: Be prepared.

Technical Info

All graphs were created with Mathematica 8, one of my favorite pieces of software. It's a tremendously powerful package and, though it does have a bit of a learning curve, the help system is excellent at giving lots of examples.

There's a lot more that could, and probably should, be done with this data: time-based analysis, looking for correlations, leveraging more of the data (e.g. tsunamis, impact of earthquakes, etc.).

The graphs above are pretty simple. Mathematica can create much more sophisticated ones. Here's 3D version of the Japan-area earthquakes.

 
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Paul Clip
tag:artscience.cyberclip.com,2013:Post/341970 2011-02-20T00:39:05Z 2013-10-08T16:34:44Z Real Life Version of Escher's Waterfall Escher's my favorite artist, I just love the way he married art and math to create wonderful illusions. One of his famous works is Waterfall:

So I was impressed and excited to see this amazing reconstruction...

Equally fun are people's comments trying to guess how mcwolles built this.

The Waterfall is easier to build inside a computer...

/hat tip to Laughing Squid
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Paul Clip
tag:artscience.cyberclip.com,2013:Post/341973 2011-01-07T22:28:54Z 2013-10-08T16:34:44Z A Puzzle To Die For Came across this very cool puzzle at Brisbane's Museum of Science: they'd cut a die into nine pieces and it was up to you to put it back together. Being a big fan of polyhedra the puzzle instantly appealed to me, but I also like the fact that it forced people to think of how a die is constructed. For example, which faces are opposite which? Given there were only nine pieces it wasn't hard putting it back together again but Katrine and I had fun solving the puzzle nonetheless.

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tag:artscience.cyberclip.com,2013:Post/341978 2010-12-28T14:19:28Z 2013-10-08T16:34:44Z Mathematica, Pi, Mandelbrot, OpenCL, and More As usual I was looking for something completely different (pages devoted to the game Hex, but that's another story) and came across this intriguing finding by Dave Brol.

Dave was investigating whether the "neck" of the Mandelbrot set was infinitely thin, and in the process realized that pi was showing up in the number of iterations he was performing. What the?!

Curious, I whipped up a quick Mathematica function to test his observation (and as another excuse to play with Mathematica :-)

First I define Mandelbrot's function, here called mandelIter. Nice, I can actually use complex numbers in Mathematica. This function is usually written to include a stopping condition based on a max number of iterations. I chose to omit it here since the points we're investigating are supposed to eventually escape the set.

The point in question is (-0.75 + x i) where x -> 0. The table shows what occurs for ever more "precise" (i.e. closer to 0) values of x. The right hand column lists pairs of numbers. The left one is the time it took to execute mandelIter in seconds. The right one is the number of iterations before escaping the set.

Notice anything special regarding those iteration counts? Yup, we're re-discovering pi.

The other thing you may have noticed is that it's going to take a long long time to calculate pi this way. Each subsequent finding takes 10 times longer to compute.

Naturally, I wondered how to make mandelIter faster.

Mathematica includes a Compile[] function, which basically turns a function into an optimized version, though it's still in pcode, not machine code.

My first attempt was just to compile the Abs[] and z = z^2 + c functions, but this resulted in trivial speed improvements. The solution was to let Mathematica compile the whole function.

Much faster! Unfortunately, it runs into problems and never returns when I go to 10^-8:

My assumption is that compilation uses machine-level precision instead of the arbitrary precision that's Mathematica's default. Since we're squaring very small numbers, we're rapidly exceeding this precision. It's pretty cool that Mathematica realizes this and switches back.

Still, I was hoping for more.

One of the features new to Mathematica 8 is the ability to leverage your computer's GPU, i.e. your graphics processor. Modern graphics chips actually have a massive amount of computing power and are highly parallelized. You typically program these chips with either CUDA (which is NVidia specific) or OpenCL (an industry standard). As of Snow Leopard, Apple includes OpenCL drivers with its Macs.

Re-implementing the algorithm was pretty easy: you're writing C code with an OpenCL flavor. Previous versions calculated each row sequentially. This version calculates them in parallel. After all, there are 48 (!) cores on my laptop's GPU :-)

mandelCLsrc takes 5 arguments: two input vectors, one output vector, the length of the vectors, and the max iterations we want to compute. Then I compile it with OpenCLFunctionLoad[]. Notice the "8" parameter: that means we want to leverage 8 cores to process this function.

If you're doing any OpenCL programming, definitely use OpenCLFunctionLoad[]'s ShellOutputFunction -> Print option, it'll help you find errors. While I'm on the subject: if Mathematica stops compiling OpenCL functions or behaves strangely, just restart it. OpenCL gives you very low level memory access, it's not hard to corrupt things.

So how did this solution fare? Disappointingly :-(

I define my two input vectors, xt and yt, and one output vector nt. Sadly, while the first few iterations are promising, it appears that we quickly run out of precision here too. With one thousandth as input, the function should return in roughly 31,420 iterations. Clearly it does not.

I tried switching from floats to double, even though my chip apparently doesn't support them, this was no help.

Where to go from here? Mathematica's ParallelEvaluation[] function, which splits jobs across multiple Mathematica kernel, won't really help. Another option could be to switch to integer arithmetic for Mandelbrot's function though I'm not convinced this would improve Mathematica's speed. There's always a chance that I may have done something wrong in my OpenCL code, or that further optimizations are possible in the Mathematica code. A final option that I couldn't get to work was running the OpenCL code on my CPU instead of my GPU. This is theoretically possible and would leverage the CPU's ability to handle higher precision doubles.

Implementation in a different language may be the ultimate solution. C, C++, or Java make most sense to me given they have excellent optimizing compilers but, just for kicks, I picked one of my favorite programming languages: ruby.

Here are the results running on ruby 1.9.2 (ruby 1.8.7 is about 1/2 as fast):

1,          3,        2.0e-05
1/10,       33,       4.4e-05
1/100,      315,      0.000312
1/1000,     3143,     0.003746
1/10000,    31417,    0.031083
1/100000,   314160,   0.300709
1/1000000,  3141593,  3.02154
1/10000000, 31415927, 30.608127

To my surprise modern ruby is significantly slower than compiled Mathematica at this task. I would have expected ruby to fare better.

Interestingly, both ruby and Mathematica bog down significantly when computing 10^-8. They don't return in a meaningful timeframe, i.e. I killed both calculations after 30 minutes. Presumably this is the point where both switch to arbitrary length floating point routines in software.
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Paul Clip
tag:artscience.cyberclip.com,2013:Post/341991 2010-12-22T01:56:00Z 2015-02-18T01:29:27Z You have 100 feet of fencing, what shape will give you the biggest field?

That's the question I asked my sons today. It's an interesting problem but to make it a little more interesting and apropos given we're in Australia, I asked them: "If you were a crocodile farmer and only had a 100 foot long fence, what shape would you arrange it in to have the most space for your reptiles?"

I got many different answers: Square! Pentagon! Circle! Oval! Most people assumed square was the right answer but didn't all want to pick the same shape.

So we decided to start with a more general square: a rectangle. Thanks to a length of rope (representing our 100 foot fence) we were able to prototype various areas. A square seemed best. Was that so?

We turned to Mathematica (I just bought the newly released version 8, love it) for confirmation:

The maximum area is achieved when a side is equal to 25 feet. In other words: a square gives us the max area of 625 ft^2.

What about other shapes? Mathematica to the rescue!

A triangle makes poor use of fencing, the square is an improvement but clearly the more sides our fence has, the greater the area... The optimal shape must be a circle!

Though they saw that the circular field's area was greater than the square field's, this still didn't hit home until I plotted both.

Circles rock!
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Paul Clip
tag:artscience.cyberclip.com,2013:Post/342008 2010-11-04T07:09:00Z 2013-10-08T16:34:45Z Fractals without a Computer

We often think of fractals as computer-generated but they're all around us, from coastlines to trees. This video shows how a few ingenious people generate some of the most famous "computer fractals" entirely by analog means: a few cameras and a feedback loop. Watch closely and you'll see Julia sets, Sierpinski gaskets, and more! Another way to remember Benoit Mandelbrot's recent passing.

More details here.
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Paul Clip
tag:artscience.cyberclip.com,2013:Post/342022 2010-10-07T21:11:03Z 2013-10-08T16:34:45Z Venice Mosaics
San Marco's Cathedral is full of gorgeous mosaics. Indeed, its domes are covered in them. Many are gorgeous, vibrant, and some so beautifully rendered as to look almost like paintings. This is especially true of the first picture below: Saint Marc, patron saint of Venice.

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Paul Clip
tag:artscience.cyberclip.com,2013:Post/342025 2010-10-05T10:41:17Z 2013-10-08T16:34:45Z Exploring the Science of Sounds with Wolfram|Alpha and Audacity Saturday evening at bedtime, our 9 year old son Daniel told me: "Dad, I saw this program on TV. And it was two people talking, just like you and me. It was about sound and sound waves. I want to learn everything about sound. Will you teach me?"

Music to my ears :-)

So I pulled out two tools of choice: Wolfram|Alpha and Audacity. W|A is a computational search engine and, as my boys used to say, hecka useful (using "hecka" doesn't appear to be cool anymore BTW). Audacity is a swiss army knife audio recorder, editor, analyzer, and effects machine available for Mac, PC, and Linux.

Our "lesson" went back and forth between these two tools, with the real world intruding once in a while...

First off: What is sound? We talked about compression & expansion of air, how our ears work, and how sound does and doesn't travel through different mediums (like water & space).

Then I introduced the concept of Hertz. Here we used Audacity to generate an ever increasing tone from 1 to 300 Hz over 60 seconds. When do we start hearing sound? (On my Macbook Pro's speakers, around 100Hz).

Once Hertz were understood (and we'd talked about the cool low Hz rumbles our subwoofer generates) it was time to break out Wolfram|Alpha, which has some useful sound generation features.

We initially played a simple 440Hz tone (which is the standard tuning for guitars and probably other instruments too).

I'll spare you the lengthy and enjoyable experimentation that followed on W|A :-) We tried many things: different tonescombining tones, combining prime tones, even combining lots of tones and wondering whether a small change in one of the tones was still noticeable.

At this point I decided we needed a real time spectrum analyzer (Audacity can do the job, but it isn't real time). In the old (OK, old old :-) days I'd have used the one on my NeXTstation, very useful in tuning my guitar, but today I found AudioXplorer instead. Pity it's discontinued, but fortunately it's free, open sourced... and for OS X (I'm sure there are many equivalents on Windows).

Here's that 440Hz tone again...

With AudioXplorer in hand, we headed to the piano and "looked" at lots of different frequencies. It was fun to watch Daniel's interest as he played ascending notes and saw the sonogram showing a staircase pattern. (More experimentation ensued).

Next we were back to Audacity: recording Daniel's voice and applying many fun effects to it. Echo, tempo change, reverb, and more. 

Finally, back to Wolfram|Alpha one last time: experimenting with DTMF. I had Daniel press one of the keys on a phone, then we used W|A to reproduce it. Here's the tone for the "1" key. Unfortunately W|A sometimes introduces annoying clicks when generating the sounds, however you can use AudioXplorer instead, which gives correct output.

Once we'd understood how DTMF tones work and how to generate them, we used Audacity to generate a whole phone number and I showed Daniel the trick I'd promised him earlier: how to call someone without hitting any keys on the phone's keypad. We held the handset's microphone up to the laptop's speaker, press play, and... Magic! The call went through! 

All in all we had great fun with these tools. Daniel and I learned a lot together. What does he want to do now?

GarageBand! :-)
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Paul Clip
tag:artscience.cyberclip.com,2013:Post/342044 2010-09-17T11:23:52Z 2013-10-08T16:34:45Z La Belle Geode
La Geode is a fascinating structure in Paris' Cite des Sciences. It's a 36 meter diameter IMAX movie theater built in 1985 and has the only 12.1 surround sound system in the world (can't say I noticed). More impressive are the ~6,500 triangles that coat the sphere: they are placed in groups of 4 with a tolerance of within 1/10 millimeter. No triangles actually touch each other to allow for expansion due to temperature variations. Best of all? The reflections!

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Paul Clip
tag:artscience.cyberclip.com,2013:Post/342060 2010-09-16T10:16:51Z 2013-10-08T16:34:45Z Refraction with Honey
Interesting "discovery" the other day. We noticed that a spoon in a jar of honey looked much bigger than expected, certainly more than water. The answer? The refractive index of honey is ~1.5 vs. ~ 1.3 for water. 

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Paul Clip
tag:artscience.cyberclip.com,2013:Post/342074 2010-09-05T20:03:01Z 2013-10-08T16:34:46Z Alpine Emerald Grasshopper
really like insects: they're beautiful, incredibly varied, and fascinating. Here's a lovely emerald grasshopper that Katrine shot with Alexander's help.

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Paul Clip
tag:artscience.cyberclip.com,2013:Post/342090 2010-08-28T00:18:46Z 2014-08-23T08:19:47Z How to beat your friends at dice! This is a great video on non-transitive dice. The basic idea is to design 3 dice that, no matter which die your opponent picks, you can beat them in the long run.

Even more interesting are the other properties that the presenters build on top of this. Watch the video or read the article and you'll see.

/via @azaaza
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Paul Clip
tag:artscience.cyberclip.com,2013:Post/342093 2010-08-08T16:43:00Z 2015-08-11T09:45:29Z Nature by Numbers: A Beautiful Combination

Fibonacci numbers, the golden mean, seashells, and sunflowers... This short movie is a gorgeous celebration of it all.

/via @igrigorik
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Paul Clip
tag:artscience.cyberclip.com,2013:Post/342103 2010-08-06T06:48:26Z 2013-10-08T16:34:46Z Root Bench
Saw this bench by a Swiss chalet. Love the way the artist used gnarled roots to build it.

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Paul Clip
tag:artscience.cyberclip.com,2013:Post/342112 2010-07-24T16:48:55Z 2013-10-08T16:34:46Z "Don't play with matches!" I'm glad Caleb Charland wasn't listening. His photographies are a delightful mix of art and science, all the more impressive for having been taken "au naturel". I.e. no photoshop involved.

Here's one of my favorites. Simple, well executed, ingenious. The others are well worth watching.

(via @JadAbumrad)
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Paul Clip
tag:artscience.cyberclip.com,2013:Post/341925 2010-07-16T00:51:35Z 2013-10-08T16:34:43Z The Addiator: Ingenious Mechanical Pocket Calculator Growing up I loved to play with my father's Addiator, a small mechanical calculator that could easily add and subtract numbers... It wasn't a PDA but a PMA: Pocket Mechanical Assistant :-)

It's based on a very clever idea: parallel tooth metal strips had numbers from 0 to 9 printed on them, as well as a "flag" (in my addiator's case a red arrow) to indicate that you need to carry the operation over to the next column. This mechanism was originally invented in the late 1800s by a Frenchman named Louis Troncet. I've included a picture from his 1889 patent that shows you the inner working of the device. You can view a number of his patents here.

The German company Addiator manufactured devices like mine from the 1920s to the early 1980s when digital calculators finally made them obsolete. Many other companies made them too, and not just to add / subtract either

This short video shows you how simple and effective Troncet's invention was.

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Paul Clip
tag:artscience.cyberclip.com,2013:Post/341926 2010-07-06T15:31:05Z 2013-10-08T16:34:43Z Clash of Exterior Designs: Past and Future Though a little shocking at first, I think the juxtaposition of modern and traditional facades works pretty well in these two houses from Sion, Switzerland.

I assume that a glass (?) facade might have been a cheaper option than a full-on renovation. Wonder what the neighbors think of it? I do like the color coordinated fire hydrant in the lower left corner. Nice touch :-)

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Paul Clip
tag:artscience.cyberclip.com,2013:Post/341927 2010-07-04T17:13:27Z 2013-10-08T16:34:44Z Great Articles on the Beauty of Maths Earlier this year the New York Times published a great series of articles on mathematics by Cornell math professor Steven Strogatz.

Strogatz does a wonderful job sharing the wonders of maths for the lay person, starting from simple counting and finishing up with more complex topics like integration, probabilities, and some of David Hilbert's work.

Here's a beautiful example from an early column: Rock Groups. The question posed is why is the sum of consecutive odd numbers always a perfect square?
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25

At first glance this is an interesting observation but hard to understand... Until you look at these equations graphically:

And then it makes perfect sense!

Another favorite focuses on limits and gives an elegant application to finding the area of a circle.

A series well worth reading and, if you have any, sharing with your children.
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Paul Clip
tag:artscience.cyberclip.com,2013:Post/341929 2010-06-28T14:27:47Z 2013-10-08T16:34:44Z Golden Scarab, Golden Hair I love insects. Many are a beautiful blend of art and science: amazing miniaturization packaged as a work of art. This golden scarab is a perfect example. No wonder the Egyptians revered them. Sadly I only got this one shot in Katrine's hair before it flew away.
Unfortunately I have no idea what kind of scarab it is, or even if it qualifies as a scarab instead of a beetle. I think it deserves the name though!

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Paul Clip