Some WW2 or Cold-War-Era Aerial Surveillance Cameras


(Think U2 spyplanes.. )

Hopewell Observatory has three WW2 or Cold-War aerial spy camera optical tube assemblies, including a relatively famous Fairchild K-38. No film holders, though. And no spy planes. The lenses are in good condition, and the shutters seem to work fine.

We would like to give them away to someone who wants and appreciates them, and can put them to good use. Does anybody know someone who would be interested?

They’ve been sitting unused in our clubhouse for over 20 years. Take one, take two, take all of them, we want them gone.

We are located in the DC / Northern Virginia area. Nearby pickup is best. Anybody who wants them shipped elsewhere would obviously need to pay for packaging and shipping.

Here are some photos.

This one is labeled K-38, has a special, delicate, fluorite lens in front, and is stamped with the label 10-10-57 – perhaps a date. The shoe is for scale.


The next two have tape measures and shoes for scale.


Let me know (a comment will work) if you are interested.

Can Mathematicians be Replaced by Computers?


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The short answer is, certainly not yet.

Can they ever be? From reading this article and my own experience with the geometry-proving-and-construction software called Geometrix, written by my friend and colleague Jacques Gressier, I am not sure it’s possible at all.

Here is an interesting article that I’m copying and pasting from Jerry Becker at SIU, who got it from Quanta:

From Quanta, Thursday, August 27, 2020. SEE
How Close Are Computers to Automating Mathematical Reasoning?
AI tools are shaping next-generation theorem provers, and with them the relationship between math and machine.
By Stephen Ornes
In the 1970s, the late mathematician Paul Cohen, the only person to ever win a Fields Medal for work in mathematical logic, reportedly made a sweeping prediction that continues to excite and irritate mathematicians – that “at some unspecified future time, mathematicians would be replaced by computers.” Cohen, legendary for his daring methods in set theory, predicted that all of mathematics could be automated, including the writing of proofs.

A proof is a step-by-step logical argument that verifies the truth of a conjecture, or a mathematical proposition. (Once it’s proved, a conjecture becomes a theorem.) It both establishes the validity of a statement and explains why it’s true. A proof is strange, though. It’s abstract and untethered to material experience. “They’re this crazy contact between an imaginary, nonphysical world and biologically evolved creatures,” said the cognitive scientist Simon DeDeo of Carnegie Mellon University, who studies mathematical certainty by analyzing the structure of proofs. “We did not evolve to do this.”

Computers are useful for big calculations, but proofs require something different. Conjectures arise from inductive reasoning – a kind of intuition about an interesting problem – and proofs generally follow deductive, step-by-step logic. They often require complicated creative thinking as well as the more laborious work of filling in the gaps, and machines can’t achieve this combination.

Computerized theorem provers can be broken down into two categories. Automated theorem provers, or ATPs, typically use brute-force methods to crunch through big calculations. Interactive theorem provers, or ITPs, act as proof assistants that can verify the accuracy of an argument and check existing proofs for errors. But these two strategies, even when combined (as is the case with newer theorem provers), don’t add up to automated reasoning.

SIDEBAR PHOTO:  Simon DeDeo of Carnegie Mellon helped show that people and machines seem to construct mathematical proofs in similar ways. Courtesy of Simon DeDeo
Plus, the tools haven’t been met with open arms, and the majority of mathematicians don’t use or welcome them. “They’re very controversial for mathematicians,” DeDeo said. “Most of them don’t like the idea.”

A formidable open challenge in the field asks how much proof-making can actually be automated: Can a system generate an interesting conjecture and prove it in a way that people understand? A slew of recent advances from labs around the world suggests ways that artificial intelligence tools may answer that question. Josef Urban at the Czech Institute of Informatics, Robotics and Cybernetics in Prague is exploring a variety of approaches that use machine learning to boost the efficiency and performance of existing provers. In July, his group reported a set of original conjectures and proofs generated and verified by machines. And in June, a group at Google Research led by Christian Szegedy posted recent results from efforts to harness the strengths of natural language processing to make computer proofs more human-seeming in structure and explanation.

Some mathematicians see theorem provers as a potentially game-changing tool for training undergraduates in proof writing. Others say that getting computers to write proofs is unnecessary for advancing mathematics and probably impossible. But a system that can predict a useful conjecture and prove a new theorem will achieve something new –  some machine version of understanding, Szegedy said. And that suggests the possibility of automating reason itself.
Useful Machines

Mathematicians, logicians and philosophers have long argued over what part of creating proofs is fundamentally human, and debates about mechanized mathematics continue today, especially in the deep valleys connecting computer science and pure mathematics.

For computer scientists, theorem provers are not controversial. They offer a rigorous way to verify that a program works, and arguments about intuition and creativity are less important than finding an efficient way to solve a problem. At the Massachusetts Institute of Technology, for example, the computer scientist Adam Chlipala has designed theorem-proving tools that generate cryptographic algorithms – traditionally written by humans – to safeguard internet transactions. Already, his group’s code is used for the majority of the communication on Google’s Chrome browser.

SIDEBAR PHOTO: Emily Riehl of Johns Hopkins University uses theorem provers in teaching students and proof assistants in her own work. “Using a proof assistant has changed the way I think about writing proofs,” she said. Will Kirk/Johns Hopkins University
“You can take any kind of mathematical argument and code it with one tool, and connect your arguments together to create proofs of security,” Chlipala said.

In math, theorem provers have helped produce complicated, calculation-heavy proofs that otherwise would have occupied hundreds of years of mathematicians’ lives. The Kepler conjecture, which describes the best way to stack spheres (or, historically, oranges or cannonballs), offers a telling example. In 1998, Thomas Hales, together with his student Sam Ferguson, completed a proof using a variety of computerized math techniques. The result was so cumbersome – the results took up 3 gigabytes – that 12 mathematicians analyzed it for years before announcing they were 99% certain it was correct.

The Kepler conjecture isn’t the only famous question to be solved by machines. The four-color theorem, which says you only need four hues to color any two-dimensional map so that no two adjoining regions share a color, was settled in 1977 by mathematicians using a computer program that churned through five-colored maps to show they could all be reduced to four. And in 2016, a trio of mathematicians used a computer program to prove a longstanding open challenge called the Boolean Pythagorean triples problem, but the initial version of the proof was 200 terabytes in size. With a high-speed internet connection, a person could download it in a little over three weeks.

Complicated Feelings
These examples are often trumpeted as successes, but they’ve also added to the debate. The computer code proving the four-color theorem, which was settled more than 40 years ago, was impossible for humans to check on their own. “Mathematicians have been arguing ever since whether or not it’s a proof,” said the mathematician Michael Harris of Columbia University.
SIDEBAR PHOTO:  Many mathematicians, like Columbia University’s Michael Harris, disagree with the idea that computerized theorem provers are necessary – or that they’ll make human mathematicians obsolete. Béatrice Antolin
Another gripe is that if they want to use theorem provers, mathematicians must first learn to code and then figure out how to express their problem in computer-friendly language – activities that detract from the act of doing math. “By the time I’ve reframed my question into a form that could fit into this technology, I would have solved the problem myself,” Harris said.

Many just don’t see a need for theorem solvers in their work. “They have a system, and it’s pencil and paper, and it works,” said Kevin Buzzard, a mathematician at Imperial College London who three years ago pivoted his work from pure math to focus on theorem provers and formal proofs. “Computers have done amazing calculations for us, but they have never solved a hard problem on their own,” he said. “Until they do, mathematicians aren’t going to be buying into this stuff.”

But Buzzard and others think maybe they should. For one thing, “computer proofs may not be as alien as we think,” DeDeo said. Recently, together with Scott Viteri, a computer scientist now at Stanford University, he reverse-engineered a handful of famous canonical proofs (including one from Euclid’s Elements) and dozens of machine-generated proofs, written using a theorem prover called Coq, to look for commonalities. They found that the networked structure of machine proofs was remarkably similar to the structure of proofs made by people. That shared trait, he said, may help researchers find a way to get proof assistants to, in some sense, explain themselves.
“Machine proofs may not be as mysterious as they appear,” DeDeo said.

Others say theorem provers can be useful teaching tools, in both computer science and mathematics. At Johns Hopkins University, the mathematician Emily Riehl has developed courses in which students write proofs using a theorem prover. “It forces you to be very organized and think clearly,” she said. “Students who write proofs for the first time can have trouble knowing what they need and understanding the logical structure.”

SIDEBAR:  By the time I’ve reframed my question into a form that could fit into this technology, I would have solved the problem myself.  Michael Harris, Columbia University
Riehl also says that she’s been increasingly using theorem provers in her own work. “It’s not necessarily something you have to use all the time, and will never substitute for scribbling on a piece of paper,” she said, “but using a proof assistant has changed the way I think about writing proofs.”

Theorem provers also offer a way to keep the field honest. In 1999, the Russian American mathematician Vladimir Voevodsky discovered an error in one of his proofs. From then until his death in 2017, he was a vocal proponent of using computers to check proofs. Hales said that he and Ferguson found hundreds of errors in their original proof when they checked it with computers. Even the very first proposition in Euclid’s Elements isn’t perfect. If a machine can help mathematicians avoid such mistakes, why not take advantage of it? (The practical objection, justified or not, is the one suggested by Harris: If mathematicians have to spend their time formalizing math to be understood by a computer, that’s time they’re not spending doing new math.)

But Timothy Gowers, a mathematician and Fields medalist at the University of Cambridge, wants to go even further: He envisions a future in which theorem provers replace human referees at major journals. “I can see it becoming standard practice that if you want your paper to be accepted, you have to get it past an automatic checker,” he said.

But before computers can universally check or even devise proofs, researchers first have to clear a significant hurdle: the communication barrier between the language of humans and the language of computers.

Today’s theorem provers weren’t designed to be mathematician-friendly. ATPs, the first type, are generally used to check if a statement is correct, often by testing possible cases. Ask an ATP to verify that a person can drive from Miami to Seattle, for example, and it might search all cities connected by roads leading away from Miami and eventually finding a city with a road leading into Seattle.

SIDEBAR PHOTO: Not every mathematician hates theorem provers. Timothy Gowers, of the University of Cambridge, thinks they may one day replace human reviewers at mathematical journals. The Abel Prize
With an ATP, a programmer can code in all the rules, or axioms, and then ask if a particular conjecture follows those rules. The computer then does all the work. “You just type in the conjecture you want to prove, and you hope you get an answer,” said Daniel Huang, a computer scientist who recently left the University of California, Berkeley, to work at a startup.

But here’s the rub: What an ATP doesn’t do is explain its work. All that calculating happens within the machine, and to human eyes it would look like a long string of 0s and 1s. Huang said it’s impossible to scan the proof and follow the reasoning, because it looks like a pile of random data. “No human will ever look at that proof and be able to say, ‘I get it,'” he said.

ITPs, the second category, have vast data sets containing up to tens of thousands of theorems and proofs, which they can scan to verify that a proof is accurate. Unlike ATPs, which operate in a kind of black box and just spit out an answer, ITPs require human interaction and even guidance along the way, so they’re not as inaccessible. “A human could sit down and understand what the proof-level techniques are,” said Huang. (These are the kinds of machine proofs DeDeo and Viteri studied.)
ITPs have become increasingly popular in recent years. In 2017, the trio behind the Boolean Pythagorean triples problem used Coq, an ITP, to create and verify a formal version of their proof; in 2005 Georges Gonthier at Microsoft Research Cambridge used Coq to formalize the four-color theorem. Hales also used ITPs called HOL Light and Isabelle on the formal proof of the Kepler conjecture. (“HOL” stands for “higher-order logic.”)

Efforts at the forefront of the field today aim to blend learning with reasoning. They often combine ATPs with ITPs and also integrate machine learning tools to improve the efficiency of both. They envision ATP/ITP programs that can use deductive reasoning – and even communicate mathematical ideas – the same way people do, or at least in similar ways.

The Limits of Reason

Josef Urban thinks that the marriage of deductive and inductive reasoning required for proofs can be achieved through this kind of combined approach. His group has built theorem provers guided by machine learning tools, which allow computers to learn on their own through experience. Over the last few years, they’ve explored the use of neural networks – layers of computations that help machines process information through a rough approximation of our brain’s neuronal activity. In July, his group reported on new conjectures generated by a neural network trained on theorem-proving data.

Urban was partially inspired by Andrej Karpathy, who a few years ago trained a neural network to generate mathematical-looking nonsense that looked legitimate to nonexperts. Urban didn’t want nonsense, though – he and his group instead designed their own tool to find new proofs after training on millions of theorems. Then they used the network to generate new conjectures and checked the validity of those conjectures using an ATP called E.

The network proposed more than 50,000 new formulas, though tens of thousands were duplicates. “It seems that we are not yet capable of proving the more interesting conjectures,” Urban said.

SIDEBAR: [Machines] will learn how to do their own prompts.  Timothy Gowers, University of Cambridge
Szegedy at Google Research sees the challenge of automating reasoning in computer proofs as a subset of a much bigger field: natural language processing, which involves pattern recognition in the usage of words and sentences. (Pattern recognition is also the driving idea behind computer vision, the object of Szegedy’s previous project at Google.) Like other groups, his team wants theorem provers that can find and explain useful proofs.

Inspired by the rapid development of AI tools like AlphaZero – the DeepMind program that can defeat humans at chess, Go and shogi – Szegedy’s group wants to capitalize on recent advances in language recognition to write proofs. Language models, he said, can demonstrate surprisingly solid mathematical reasoning.

His group at Google Research recently described a way to use language models – which often use neural networks – to generate new proofs. After training the model to recognize a kind of treelike structure in theorems that are known to be true, they ran a kind of free-form experiment, simply asking the network to generate and prove a theorem without any further guidance. Of the thousands of generated conjectures, about 13% were both provable and new (meaning they didn’t duplicate other theorems in the database). The experiment, he said, suggests that the neural net could teach itself a kind of understanding of what a proof looks like.

“Neural networks are able to develop an artificial style of intuition,” Szegedy said.

Of course, it’s still unclear whether these efforts will fulfill Cohen’s prophecy from over 40 years ago. Gowers has said that he thinks computers will be able to out-reason mathematicians by 2099. At first, he predicts, mathematicians will enjoy a kind of golden age, “when mathematicians do all the fun parts and computers do all the boring parts. But I think it will last a very short time.”
 Machines Beat Humans on a Reading Test. But Do They Understand?  —
 Will Computers Redefine the Roots of Math?  —
 Symbolic Mathematics Finally Yields to Neural Networks  —
After all, if the machines continue to improve, and they have access to vast amounts of data, they should become very good at doing the fun parts, too. “They will learn how to do their own prompts,” Gowers said.

Harris disagrees. He doesn’t think computer provers are necessary, or that they will inevitably “make human mathematicians obsolete.” If computer scientists are ever able to program a kind of synthetic intuition, he says, it still won’t rival that of humans. “Even if computers understand, they don’t understand in a human way.”


Logs and Plagues

I just did the math in two ways: if each person infects 5 people who have never been infected, it only takes a bit more than 14 cycles from “patient zero” (whoever that was) to infect the entire living human population.

Obviously the real progress of an epidemic isn’t that simple.

Being a retired math teacher I figured this was a perfect case for using logarithms, so I did. (For me, that’s fun!) I went like this:

I’m trying to find n such that five to the nth power equals 7.5 billion, or in math-lingo,

5^n = 7.5*10^9

One takes the logarithms of both sides, and because of the wonderful properties of logs, I get n*log(5)=9+log(7.5) which we can solve for n by dividing both sides by log(5), obtaining

n = (9+log(7.5))/log(5), after which my calculator said n was about 14.1.

But if you have a cell phone you can confirm my result much more easily by asking it work out 5^14. I think you’ll find it’s about six billion; if you try 5^15 you’ll get an enormous umber, over 30 billion, which is much too high. We have only roughly seven and a half billion humans…

A neat geometry lesson! And a rant…


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Here is some information that teachers at quite a few different levels could use* for a really interesting geometry lesson involving reflections involving two or more mirrors, placed at various angles!

Certain specific angles have very special effects, including 90, 72, 60, 45 degrees … But WHY?

This could be done with actual mirrors and a protractor, or with geometry software like Geometer’s Sketchpad or Desmos. Students could also end up making their own kaleidoscopes – either with little bits of colored plastic at the end or else with some sort of a wide-angle lens. (You can find many easy directions online for doing just that; some kits are a lot more optically perfect than others, but I don’t think I’ve even seen a kaleidoscope that had its mirrors set at any angle other than 60 degrees!)

I am reproducing a couple of the images and text that Angel Gilding provides on their website (which they set up to sell silvering kits (about which I’ve posted before, and which I am going to attempt using pretty soon)).

At 72º you see 4 complete reflections.

When two mirrors are parallel to each other, the number of reflections is infinite. Placing one mirror at a slight angle causes the reflections to curve.


Rant, in the form of a long footnote:

* assuming that the teacher are still allowed to initiate and carry out interesting projects for their students to use, and aren’t forced to follow a scripted curriculum. It would be a lot better use of computers than forcing kids to painfully walk through (and cheat, and goof off a lot) when an entire class is forced to use one of those very expensive but basically worthless highly-centralized, district-purchased computer-managed-instruction apps. God, what a waste of time – from personal experience attempting to be a volunteer community math tutor at such a school, and also from my experience as a paid or volunteer tutor in helping many many students who have had to use such programs as homework. Also when I was required to use them in my own classes, over a decade ago, I and most of my colleagues found them a waste of time. (Not all – I got officially reprimanded for telling my department chair that ‘Renaissance Math’ was either a ‘pile of crap’ or a ‘pile of shit’ to my then-department head, in the hearing of one of the APs, on a teacher-only day.

Keep in mind: I’m no Luddite! I realized early on that in math, science, and art, computers would be very, very useful. I learned how to write programs in BASIC on one of the very first time-share networks, 45 years ago. For the first ten years that my school system there was almost no decent useful software for math teachers to use with their classes unless you had AppleII computers. We had Commodore-64’s which were totally incompatible and there were very few companies (Sunburst was one) putting out any decent software for the latter. So when I saw some great ideas that would be ideal for kids to use on computers to make thinking about numbers, graphs, and equations actually fun and mentally engaging, often I would have to write them my self during whatever free time I could catch, at nights and weekends. Of course, doing this while being a daddy to 2 kids, and still trying to teach JHS math to a full load of students (100 to 150 different kids a day at Francis Junior High School) and running a school math club and later coaching soccer. (I won’t say I was a perfect person or a perfect teacher. I believe I learned to give better math explanations than most, didn’t believe that you either have a ‘m,ath gene’ or you don’t, at times had some interesting projects, and at times was very patient and clear, but had a terrible temper and often not good at defusing things. Ask my kids or my former students!) Later on, I collaborated with some French math teachers and a computer programmer to try to make an app/program called Geometrix for American geometry classes that was supposed to help kids figure out how to make all sorts of geometric constructions and then develop a proof of some property of that situation. It was a failure. I was the one writing the American version, including constructions and tasks from the text I was currently using. There was no way I could anticipate what sorts of obstacles students would find when using this program, until I had actual guinea pig students to use them with. Turns out the final crunch of writing however many hundreds of exercises took place over the summer, and no students to try them on. Figuring out hints and clues would require watching a whole bunch of kids and seeing what they were getting right or wrong. In other words, a lot of people’s full time job for a long time, maybe paying the kids as well to try it out so as to get good feedback, and so on. Maybe it could work, but it would require a lot more investment of resources that the tiny French and American companies involved could afford. We would have really needed a team of people, not just me and a single checker.

I find that none of these computer-dominated online learning programs (much less the one I worked on) can take the place of a good teacher. Being in class, listening to and communicating logically or emotionally with a number of other students and a knowledgeable adult or two, is in itself an extremely important skill  to learn. It’s also the best way to absorb new material in a way that will make sense and be added to one’s store of knowledge. That sort of group interaction is simply IMPOSSIBLE in a class where everybody is completely atomized and is on their own electronic device, engaged or not.

Without a human being trying to make sense out of the material, what I found quite consistently, in all the computerized settings, that most students absorbed nothing at all or else the wrong lessons altogether (such as, ‘if you randomly try all the multiple choice answers, you’ll eventually pick the right one and you can move on to some other stupid screen’; it doesn’t matter that all your prior choices were wrong; sometimes you get lucky and pick the right one first or second! Whee! It’s like a slot machine at a casino!).

By contrast, I found that with programs/apps/languages like Logo, Darts, Green Globs, or Geometer’s Sketchpad, with teacher guidance, students actually got engaged in the process, had fun, and learned something.

I find the canned computer “explanations” are almost always ignored by the students, and are sometimes flat-out wrong. Other times, although they may be mathematically correct, they assume either way too much or way too little, or else are just plain confusing. I have yet to detect much of any learning going on because of those programs.

In which some of the advantages of traditional Dobsonian telescopes are demonstrated …

Darwin B recently built, in nearly record time, an interesting, two-truss, tubeless, collapsible, travel-ready Newtonian scope at our DC-area telescope making workshop, using an 8″ parabolic mirror with a short focal length that he bought.

He mounted it on a commercial alt-az tripod, as you see here. It will definitely collapse and fit either in a suitcase or fit in carry-on spaces on an airplane.

Unfortunately, there are some drawbacks, as he is the first to admit:

  • The lack of any sort of light shielding is a huge problem virtually anywhere within a hundred miles of a city;
  • The ultra-cantilevered mount makes it very wiggly; images essentially never stabilize. It’s also extremely susceptible to breezes.

To quote from a recent email from him:


So let me first plead mea culpa!

1. Anywhere near DC-MD-VA, open structure telescopes are TRULY instruments of the devil! (How do I come to fully agree with you on this point? Well Tuesday morning before dawn, in the cold, I set up near work – well that 70mm mirror does an incredible job collecting light from a wide area! That also explains my difficulty at CC the other night.)
2. I am asking for “un-attainium” with my scope: big aperture, fit as 2nd item carry-on, have a good mount, and be useful locally.
3. Perhaps many people would be better off traveling with binoculars- smaller & less hassle than a scope. And would do a great (limited) job anywhere.
So what to do?
1. Fix what I can on this scope and accept limitations- fix spider, swap sides for saddle, and add a shroud. Limit to low & medium power and enjoy.
2. Since I have an 8” f/6 mirror, build a scope for around here & car travel: and not have the limitations of the other scope. I already have many/most of the parts for a design similar to your 6” f/8. Like you say, it’ll be steady, and I can crank up the power a bit for moon&planets. It just has to have the mount break down flat like an IKEA.
So, I’ll be up at the shop next Tuesday nightto drill holes to flip saddle. I should have other things done or started.
So – now you have a pretty ringing endorsement for your thoughts.
I can compare my current effort to a beach house or a boat; wouldn’t want to live there year-round. BUT they can be fun, within limits.

Mining at the Observatory (sort of…)


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We have been concerned with the status of some of the columns that are part of the roll-off-roof of the Hopewell Observatory, so we decided to remove a couple of courses of cinderblock to see what was inside. It turned out to be built much more sturdily than they appeared. and removing those two layers of cinderblock ended up being a much harder job than we expected. We had to build a very strong ‘crib’ to hold the upper part of the 9-foot-tall column in place while we removed the lower foot-and-a-third.

In the video, you see me using a small hand-held air-hammer with chisel to clean up the underside of the upper part of the column, so that the new solid cinderblocks can be mortared into place. The buzzing noise you hear is the air compressor.

We didn’t realize there was rebar (reinforcing iron bars) and concrete poured into most of the ‘cells’ of the 16″ by 24″ columns. Now we do.

(In the summer of 1970, between my junior and senior years, I found a job in Brooklyn working on a rodding truck for the local electric power utility, Con Edison — a hard and dirty job that made me itch constantly because of all the fiberglass dust that was scraped off the poles we used to clean out the supposedly empty, masonry, electric conduits that went from one manhole to the next. I guess I pissed off our truck crew’s supervisor, so the very day that I was about to quit to go back to college, I was told that I was being transferred to a jack-hammer crew, where I probably would have gone deaf. This woulda been me, except I quit)

Image result for jackhammering


After that was done, I trimmed some of the trees to the west. Constant struggle with the shrubbery!


More about spray-coating astronomical mirrors with silver!

Here is a batch of articles and links concerning the spray-on process for making astronomical mirrors reflective using protected silver solutions.

Long ago, I translated Foucault’s monograph on making paraboloidal, silvered astronomical mirrors. Part of his article described the process that he and Steinheil developed for silvering, which involved using silver nitrate solutions and various other reagents. It looked quite tricky, and also required further polishing! Plus, our telescope making workshop here in Washington DC had a Navy surplus vacuum chamber that was (and still is) quite effective at putting on good-quality, inexpensive aluminum coatings for any mirror up to 12.5″ diameter.

However, I and a couple of other ATMers (Bill R and Oscar O) are working on mirrors in the 16 to 18 inch range, and they simply won’t fit. So I was quite intrigued to watch how Peter Pekurar and some other folks coated a couple of rather large mirrors right in front of a small crowd of onlookers in a tent at this summer’s Stellafane.

I have a few videos on my webpage (here).

There is also an article on the process in the January 2020 Sky and Telescope, and a webpage (here) on the topic run by Pekurar and Howard Banich and others.

Not to mention a bunch of posts on Cloudy Nights (here) and a nice PDF explaining it all, (here).

What is really, really amazing is that the webpage by Pekurar and Banich also has interferograms showing that the overcoating has absolutely no effect on the sub-microscopic, geometrical figure of the mirror! Unfortunately, it’s only effective against chemical attack, not against dirty fingers or scratches. They also did some careful experiments on reflectivity at various wavelengths with various treatments of the surface.

A couple of local ATMers and at least one professional at Goddard Space Flight Center have told me about their experiments with the process; they found that it is easy to mess up if you aren’t stringently clean and also easy to waste materials.

James Tanton : what is K-12 math?

Jim Tanton is a very deep thinker and communicator about many aspects of mathematics. He recently was in residence in the DC area for a few years and was a mentor at Math for America – DC (based at the Carnegie Institution for Science), where I attended several of his highly entertaining and inspiring talks for new and experienced DC secondary Math teachers.

This article by him goes into what mathematics is all about, and how we teach [a part] of that in school. Here is the link: