Flexagon, Folding Paper Machines

Flexagon, Folding Paper Machines

This is my flexagon. It is a tetra-tetra flexagon. Or has four faces and four sides. By folding, and sometimes pinching and flexing, these little paper devices you can check change numbers or colors or patterns. Numbers or colors or patterns well up from seemingly flat dimensions in an (almost) never ending succession like a mobius-strip come alive! Although instead of creating a 3D shape from a flat piece of paper and then making it 1D this mobius-strip "snake", the flexagon takes a flat piece of paper and cycles of into the realm of 3D momentarily and then end with a 1D/2D changed first world.

Each flex, pinch, fold or flip can change the image, pattern or number by a little or a lot. Flip, meow, flip!


In 1939 Arthur Stone, Bryant Tuckerman, Richard Feynman and John Tukey published a paper of their mathematical findings after Arthur Stone discovered the hexa-flexagon. Flexagons can have as few as three sides up to an infinite number: whatever your brain can come up with, provided you have a big enough  piece of paper.

Here is my humble collection of flexagons. There are various shapes, which necessitate different maneuvers-not just flipping like the first video.

Pinch, flex, pinch, flex...

This last hexa-hexa-flexagon has six sides and six faces, so many in fact that I stopped flexing it at just the 5th face/side...I just got lost between dimensions and couldn't find the sixth.

I'm stuck in dimensions too-meow!

Jacob's Ladder Toy

Something very similar to a paper flexagon is this Jacob's Ladder Toy I made from some La Florentine candy boxes that my boss gave me.

There are tons of great instructions online on how to make these. My two tips: don't make the ribbon attachments too tight; and use heavy boxes. I ate the candy, so my boxes are too light and don't flip as fast as they could.

Although it's "automatic" and quite impressive looking in action, the Jacob's Ladder Toy is much less sophisticated than a flexagon. Add we've seen previously, a flexagon can have many phases it can cycle through. My Jacob's Ladder Toy can only flip the boxes upside down or right-side-up.

I'll probably throw some pennies into the boxes to weight them better.

see atomic particles with your own eyes huh? Part II

So you wanna see atomic particles with your own eyes huh? Part II

Was der Fall ist, die Tatsache, ist das Bestehen von Sachverhalten.

(What is the case, the fact, is the existence of atomic facts.)

-Ludwig Wittgenstein

Wittgenstein was a fascinating weirdo, but in the second proposition to his only (seventy-five page long) contribution to the world of philosophy and logic (the Tractatus Logico Philosophicus) he wrote the quote above. He wasn't denying art, religion, myth, but when speaking of absolute truth there are facts (atomic and otherwise) and then there is everything else.  Stay in there, it gets less boring real soon...

Basically he went against metaphysics (sort of) and Plato, et al and piled on a bunch of stuff about words and the world and truth functions (like actual math functions but with words) and examining the world using language. If any of the above grabs you: search "atomic sentence" and there's a whole world of true/false wordy-math formula semantic whatever. Anyway, Wittgenstein argued for unalterable objects/forms in direct opposition to eastern philosophy where Forms are ever changing, relative substances in a constant state of flux (sounds just like radioactive elements becoming different substances by adding/losing electrons and particles through decay. Think back to the last post about changing the dime's silver "form" to different isotopes 107, 108, 109 and 110. Flux. Change. Pretty much the opposite of Wittgenstein's ideas, but my quest to see atoms isn't.

Anyway, Wittgenstein claimed he solved all the problems of philosophy(!) so there's no need to continue in that vein. Here's an Atomic Facts of my own: I've seen the trails left by alpha, beta and gamma particles with my cloud chamber. This was covered in one of my previous posts. Think of the nuclear cloud chamber like standing in fog and watching bullets zing past you ripping lines through the fog. Cool-but I want more!

This is where my new toy comes in: a radioscope / sphinthariscope. A radioscope is a screen with zinc-sulfide paste smeared on it. It also has a magnifying eyepiece attached. You hold it up to alpha emitting radioactive objects to see tiny flashes of light when the alpha particles hit the zinc-sulphide crystal and form a phosphor (not to be confused with the chemical element phosphorus). It glows in the dark, well actually it scintillates. 

The word sphinthariscope comes from the Greek word for scintillate or sparkle. A sphinthariscope comes with a built in piece of radioactive material to "power" it-a radioscope is the same thing but with the radioactive material removed, along with the bottom of the device so that you can plop it on top of your own radioactive materials.

For $29 you can buy one from United Nuclear-but for about $9 you buy the activate zinc-sulfide and smear it on your own homemade scope. I opted to order one first, I'll probably make one from scratch for the fun of it in the future.

William Cooke made/discovered the first spinthariscope when using a phosphor screen to look for bits of radioactive material he spilled on the floor (been there-done that). He was lighting the screen with an even about of materials, but crawling around on the floor let him see tiny amounts of (alpha) radiation as individual flashes-not just an even glow. 

They became popular novelties in the early 1900s to 1950s: nice brass ones that people took to fancy dinner parties were the "in" thing. The famous Lone Ranger Atomic Ring was a later one that tons of kids received after making their Kix Cereal boxtops in. Then the fad waned.

So how did my radioscope work?

At first there was nothing, but after about ten minutes in a dark room I could see the alpha sparks. My eyes took time to adjust, but the zinc-sulfide coating glows in the dark for a few minutes as well.  At 6400 ISO with a fast f/1.8 lens I couldn't photograph them. Digital camera: 1, human eyeball: 1.

My uranium ore had random green flashes like looking out at a vast field and watching for fireflies to flash. There were some sideways "zingers" and bigger, smeared flashes like lightening behind clouds.

As "hot" as my uranium is to my Geiger counter (which measures beta, gamma and x-ray) there wasn't too much alpha going on. It was nice and subtle and I could have watched for hours, but I didn't like having uranium two inches from my face blasting intense radiation into my eyeball like some brain cancer inducing Medusa.

I moved my radioscope off the pile of uranium and plopped it on top of my little piece of Americium radioisotope 241 (AM-241). AM-241 spews out lots of alpha and a fair amount of gamma radiation. It's what ionizes the chamber inside many cheap smoke detectors: smoke particles block the alpha particles (they're weak) and trigger the alarm.

The verdict with Americium? Wow!! At first it was a dense, waving matrix of corruscating green dots like an old computer monitor from the 1980s that was being reflected in a wavy lake at night. Green dots pulsating, then the dots would surge and swirl like a Hindu mandala (which metaphysically symbolize the universe-Wittgenstein would not approve). 

Imagine the pattern on m my kilim rug, if the rug was hanging on a clothes line and the wind was blowing it toward and away from you in billowing ripples. Mesmerizing!

Sometimes it looked as though the dots were fruit flies or tiny gnats swarming (if gnats glowed in the dark). 

So, while the cloud chamber I built was like having bullets cut trails through fog, this radioscope is kinda like driving a fast car with your headlights off through a pitch black field and having fireflies splattering on the windshield...plus swirling and pulsing like a car wash on that windshield. I have lots of experience with glowing insects on my vehicle (still no idea what the one I took a photo of above was).

The surging and receding coruscating waves appear to be just like the magnetic ferri fluid (iron particles in liquid showing the magnetic field). The lapping waves of radiating particles being emitted in all directions (but viewed as they hit the flat bottom of the radioscope). 

In this flower photo the red tips are like the green alpha dots. By their speed, number and brightness you can infer many things (like the yellow paths). The charge of alpha particles was first investigated with a sphinthariscope, and research on the charge of electrons was furthered by its use; along with the correct form of atoms and their nucleus.

Imagine instead of black magnetic fluid outlining a magnetic field, green dots outlining a field radiation-smashed against the flat viewing window of my radioscope.

With these two easy to make/cheap to buy devices I've seen the paths of radioactive particles and partially how the radiate.

I think I can see the particles too! Meow.

My Radioactive Dime

My Radioactive Dime

Alpha particle hits aluminum.

Aluminum releases neutrons.

Neutrons hit silver dime and turns it radioactive.

Silver dime releases gamma radiation.

Alpha particles turn Aluminum-27 to phosphorus-30 (*see footnote)

This is accompanied by awesomely unique neutron radiation. 

What's so great about neutrons? They can transmute elements by sticking to them and changing them, sometimes making them radioactive. Remember: alpha, beta, gamma and x-ray radiation stops when the source is taken away. Turn off an x-ray machine and it's safe to walk into the room. Turn off the microwave oven and you can stick your hand safely inside to get out your nice warm popcorn.

Neutrons are a tad different though: they can irradiate elements. Blast neutrons at silver and you get radioactive silver! Turn off the neutrons and the silver is still radioactive (for a couple minutes at least).

Neutrons turn Silver-107 to silver-108 with a half life of about 2.5 minutes.

Neutrons turn Silver-109 to silver-110 with a half life of about 25 seconds.

Each of these reactions is followed by a lovely blast of easily detectable gamma radiation.

All that sounds a little fancy, how hard was it to accomplish in real life? Well, it was very simple really:

I wrapped an old silver dime in foil and left it in an alpha particle bath for a few days. I then carefully slide it next to a Geiger Counter that was shielded with paraffin wax. The baseline only mildly rose from 10cpm (clicks per minute) to 20cpm...but it did rise!

After less than two minutes it settled down to 6cpm.

10cpm was the normal background radiation in the room.

20cpm was the gamma released by the irradiated silver.

6cpm was the background radiation being partially blocked by the (now) back to normal nonradioactive silver.

Removing the silver let the rate go back to 10cpm.

These readings are very low all around, but repeatable. Back when dimes were silver the American Museum of Atomic Energy had a coin irradiating machine: pop in your dime and it would be made radioactive (for a couple minutes). Neat! But by the time you let the museum your dime transmuted back to being a boring old coin...which is good because who wants radioactive coins in their pocket?

It wasn't as momentous as when I finally got my nuclear cloud chamber working, but I did feel a little like Dr. Frankenstein and I proved my source has alpha radiation by inference since my Geiger Counter can only detect beta, gamma and x-rays...not alpha or neutron radiation; but I know both were there because of the increased gamma radiation driving up the cpm count.

Why the choice of materials used? Neutron cross sections. This tells us how interactive am element may be with neutrons. The larger the cross section the more reactive it will be. Cross sections are measured in units called "barns" which are ten to the negative twenty-fourth power in square centimeters (very, very tiny).

There are fast neutrons that can be slowed down by big scatter cross sections, and there are slowed "thermal" neutrons which can actually interact with things (like my silver dime) of those things have a big absorption cross section.

Elements/isotopes with large scatter cross sections and low mass are great neutron moderators (slower downers).

Aluminum scatters at 1.5barns and Silver absorbs at 91barns.

It's all starting to make sense isn't it!

The insanely poisonous metal Beryllium that I'm waiting for the mailman to bring me scatters at 7.6barns but only absorbs at 0.007barns...and it's the fourth element in the periodic table, meaning it had a relatively low mass. All this makes beryllium a fantastic thing to shoot alpha particles at and have it blast neutrons out in exchange!

What else is the mailman bringing me? Paraffin wax laced with Boron. Boron scatters at a respectable 5.24barns, but it absorbs at 767barns!! That will slow down lots of neutrons down to the thermal range, meaning those neutrons can actually interact better with things (my dime for instance).


This is all much more fun than using a Geiger counter that can see alpha, or a neutron detector.

Inference and implication. Pasteur said that fortune favors the prepared mind. I'll be building a zinc sulfide spinthariscope that lights up when an alpha particle hits it.

You can also modify a cheap webcam to see similar sparks when the imaging chip gets slammed into by an alpha particle.

I once had an old wristwatch. I noticed that it didn't glow steadily in the dark but twinkled. The next morning I researched it and found that it was painted with radioactive radium!

Neutrons can be detected by tubes filled with helium, lithium, boron trifluoride or a couple other pricey gases. There are also bubble dosimeters that show bubbles in a test tube where neutrons (or cosmic rays) pass through it.

Photo magnifying tubes are like night vision versions of the webcam method. Plastic scintillators light up with hit by particles-much like the spinthariscope method.

It's more fun inferring from chain reactions just like the discoverers did: a weird result that could only be explained by a previously unknown particle.

That's what you get from homemade "junk" versus fancy store bought lab equipment.

The inferences of a prepared mind. The fortune which befalls it via convoluted paths chance, accident or routine, simple experiments yeilding unexpected results that are tested instead of ignored.

Prepare your mind and fortune may follow.

*A footnote on the initial alpha and aluminum foil interactions: when the alpha particle hits the aluminum it makes phosphorus-30 plus a neutron. The phosphorus-30 decays into the usually stable silicon-30 by emitting positrons, but the silicon in this instance decays by emitting gamma radiation. The positron will likely slam into an electron and give off more gamma radiation.

Below is some great information on some older NEUTRON DETECTORS (as opposed to Geiger Counters) which you may find used online. I snagged most of this info from a 1973 report to the US Atomic Energy Commission by Alex Lorenz. If you want to read the full report, it's available as a PDF online by searching "Review of Neutron Detection Methods and Instruments".

That document has more information on each device, including the method of detection (i.e. chemical composition of scintillator crystal, etc.). It' a great document to consult if you're like many people and find just a part/tube/probe of one of these devices and want to use it with a different base/amplifier/etc.

You'll want to pay attention as to whether your Neutron detector sees fast or slow neutrons--that makes a difference in whether or not you need to use paraffin or other moderators or actually have to remove those barriers and moderators from your experiment. You don't want to slow down your neutrons with paraffin if your device can only see the fast ones and vice versa.  

DEVICE                         RANGE                       VOLTAGE

Ludlum	Fast neutrons	900v
(Model 42-2)
Eberline	slow or fast neutrons	900-1200v
(Model SPA-2)
Ludlum	1/v for thermal neutrons	900v
(Model 42-1)
Kaman	thermal & fast —	120v
(Model A-300)	0-14 MeV
Ludlum	thermal - 12 MeV	900v
(Model 42-4)
IiUdium	thermal & fast neutrons	900v
(Model 42-5)
LND	thermal neutrons	?
(Series 900)
Ortec	?	?
(System 525)
Nuclear Instruments	Linear between	?
and Chemical Corp.	10^7 and 10^12 nv
(Model 3782)
Reuter Stokes Co	1 0^15  nv	?
Reuter Stokes Cd	5X 0 ^014 nv	?
Reuter Stokes Rh	10^15 nv	?
Reuter Stokes V	10^15 nv	?
Reuter Stokes	10^10  nv	1000-1400v
(RSN-337)	(thermal)
Ludlum	thermal and fast	500-2400v
(Model 15)	neutrons
Centronics	<7.5x10^10 nv	1000v
(Type D.C. 12)
Reuter Stokes	3x10^4 to 2.5x10^5	800-900v
(RSN-17A/326/	(thermal)
330/251/327)
Reuter Stokes	10^4 to 10^11	800v
(RSN-229A)	(thermal)
Reuter Stokes	10^4 to 10^11	800v
(HSN-234A-M1)	(thermal)
Reuter Stokes	10^3 to 10^10
(RSN-15A/304/	(thermal)	100-1000v
325/332/306)
Reuter Stokes	10^3 to 10^10	200-800v
(RSN-314A)	(thermal)
Reuter Stokes	10^8 to 10^14	20-150v
(RSN-186S-M2	(thermal)
and 316S-M5)
LND	3 decades
(Series 30771)		500v
LND	5 decades	200-800v
(Series 3077)	Thermal (U235)
	or fast (U238)
(Series 3075)	Thermal	200-500v
(Series 3000,	Thermal	50-500v
Series 3050)
Centronics	9x10^3 to 9x10^7	250-500v
(PFC 16A)
Texas Nuclear	Thermal	800-1400v
(Series 9300
Texlium)
Eberline	Dose response from	1600-2000v
(PNR-4 and	thermal to 10 MeV
NRD-1)
Eberline	0.01-10^3 eV &	1300-1800v
(PNC-4)	0.2-18 MeV
Harshaw	Thermal	1700-3400v
(Model series
B3, B6, B12, B14)
Reuter Stokes	10^-3 to 10^-5	2500-3500v
(RSN-7A/7S/44/	Thermal
177S-M7/320-M2/
108S-MG)
N. Wood Model G	?	1100-2300v
Centronics	3.3x10^3 to 6x10^6	900-1100v
(Series 5EB/6)
Texas Nuclear Series 9300 Texlium	Thermal	800-1400v
LND
(Series 3000,	Thermal	50-500v
3050)
Centronics PFC 16A	9x10^3 to 9x10^7	250-500v
Centronics PFC 16B	10^11	200-400v

Phew! That was a LOT of typing! Enjoy!




Name      What is it?                                                                           Distance traveled 
                                                                                                                    through open air
Alpha       Physical particle equaling a Helium nucleus                        2-3cm

Beta          Physical particle equaling an electron                                    2-3m

Gamma    Not radioactive decay, just energy burst                                500m
                   accompanying alpha or beta radiation.
                   The same as an x-ray, but arising from
                   different sources.

Neutron     Physical particle made up of 1 up quark and                   1000s of meters
                     2 down quarks.


Name   Symbol     Makeup                           Charge         Speed       Atomic Mass Units
Alpha     α             2 protons & 2 Neutrons       + +                Slow                               4

Beta        β               1 electron                                -              Fast or Slow                1/2000

Gamma  γ                Photons/                           Neutral        Speed of light                   0
                           electromagnetic waves

Neutron n           1 up & 2 down quarks         Neutral      2.2km/S-14,000km/S         1
                                                                                                 (~5% speed of light)

Notes: 

An AMU (Atomic Mass Unit) is equal to 1/12 the mass of a Carbon-12 atom. 

Slow Neutrons are called "Thermal" and fast ones are called "Fast" neutrons. 

An alpha particle is double positive "++". 

Gamma rays are produced by atomic nuclei and x-rays are created by accelerating electrons, but they are basically the same type of wave energy. 


Lead only approximately halves the gamma/x-ray amounts. A 1/2" of lead stops about half the waves trying to get through. When beta radiation hits lead sometimes a new type of radiation is created that is more dangerous! This is Bremsstrahlung radiation (braking/deceleration). Lead barely interacts with neutron radiation, water or hydrogen-containing compounds such as common paraffin wax are much better shielding material.