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…

]]>13 Classic Mathematics Books for Lifelong Learners

Apr 26, 2018 · 8 min read

When I was a college student, I saw a list of essential math books on a blog. I promised to myself to read all those books in 10 years because there were 50 books on that list. I am still trying to finish the list. And now it is time to share some of them with you.

Mathematics is beautiful and astounding. There is a lot of joy in understanding mathematics, for instance, how the proof of Fermat’s Last Theorem or the secrets of pi, e, epsilon…

Anyway, if you passed a lot of math courses but failed to make any sense out of them during your education, those books were written for you.

Zero: The Biography of a Dangerous Idea. | Charles Seife

This is a great book that could make almost anyone love math. I found it to be quite interesting, despite my already knowing quite a bit about the topics being covered. I absolutely love Seife’s writing style. He had me laughing out loud more than a few times.

It’s hard to imagine that anyone could spend 220 pages talking about one number. While you are reading, you do not need to work through the actual with a pencil and paper. Instead the author walks you through the logic.

The history parts of this book were interesting enough to make it worth it. It covers a lot of world history. It begins with humans use of numbers and counting. I learned about Archimedes, Laplace, Newton, Euler, their contributions to math and their lives.

You can buy the book from here.

This book is a must read for math teachers. It is about how mathematics should be taught. When you start reading, you will realize that, the book is a powerful criticism on the prevalent curriculum of Mathematics in elementary, middle and high schools.

Paul Lockhart thinks that mathematics is an art, and it is much more than memorization of notations and formulas. For him, mathematics is a life-long love.

He believes that we have to stop teaching mathematics in the traditional way and we need to start using our natural curiosity to teach and learn mathematics.

It is a serious read involving actual mathematics and should be treated as a reference book and should be referenced from time to time, specially if you are in high school or college.

The author makes sure that you stay at the edge of your seat during the book, perhaps even after you’ve read the whole thing.

You can buy the book from here.

Prelude to Mathematics | W.W. Sawyer

This book is literally a classic and it is a very enjoyable read. It is about how a mathematician thinks and how to grow a mathematician.

“Prelude to mathematics” would be very meaningful for a reader who was not already familiar with such topics.

Mr. Sawyer attempts to provide an understanding of Mathematics for pretty much anyone.

You can buy the book from here.

Proofs from The Book | Aigner and Ziegler

“But to tell the truth, what they really want to prove, once in their lifetime,…”

This book contains only the “best” proofs from many different fields of mathematics. And this book is not a textbook. Ziegler and Aigner take us briefly to another world.

If you are going to read this book, you should have the knowledge of calculus and linear algebra. Otherwise, you can struggle in order to follow the proofs.

What makes it fun is that the author walks you through the most famous proofs in all of mathematics simplifying them to simple equations that you can solve in your head.

I’d heard much praise for this book and it is certainly worth it. I wish I had a book like this assigned along with the textbook during my high school math classes.

You can buy the book from here.

The Joy of x | Steven Strogatz

“The joy of x” should be beginner’s book, because this book introduces us to the wonders of mathematics very simply. It is really a brilliant introduction to mathematics.

For this book, the content is coming from a long-running blog. Each chapter is very short and about a particular feature of mathematics. Mr. Strogatz’s writing style is very engaging.

However, if you have a strong knowledge of mathematics, this book will be a easy read for you. Thus, you don’t need a PHD to enjoy this book.

You can buy the book from here.

Things to Make and Do in the Fourth Dimension | Matt Parker

This book is a nice and recommended read. It’s about “fun parts of mathematics.” The author’s writing style is so light and clear and these makes all the content understandable without a heavy background of mathematics.

]]>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)).

https://angelgilding.com/multiple-reflections/

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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.

]]>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:

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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.

]]>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)

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

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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.

]]>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:

https://medium.com/q-e-d/the-unreasonableness-of-k-12-mathematics-7fd234f25135

]]>(1) Whoever put the lens cell together last didn’t pay any attention at all to the little registration marks that the maker had carefully placed on the edges of the lenses, to show how they were supposed to be aligned with each other. I fixed that, as you see in the photo below. The reason this is probably important is that the lenses are probably not completely symmetrical around their central axes, and the maker ‘figured’ (polished away small amounts of glass) them so that if you lined them up the way he planned it, the images would be good; otherwise, they would probably not work well at all and could very well be causing the poor star test images we saw.

2. The previous assembler also put **eleven** little tape spacers around the edges, between the two pieces of glass. More is apparently not better; experts say you should have three spacers, each 120 degrees apart from the other two. Done.

3. The bottom (or ‘flint’) element is slightly smaller than the other one (the ‘crown’), so it probably shifted sideways. That alone would be enough to mess up the star tests in the way that we saw. So I wrapped two thicknesses of blue painter’s tape around the outside of the flint, and put some three cardboard shims between the edges of the ‘crown’ and the aluminum cell.

4. There were no shims at all between the flint and the aluminum ring that holds it in place underneath. This caused some small scratches on the glass, and might have been warping the glass. I put in three small shims of the same type of blue painter’s tape, lined up with the other spacers.

We will see if these improvements help. I really don’t want to haul this all the way out to Hopewell Observatory and struggle with putting it back on the mount for a star test. That was just way too much work, much more than I expected! The next test will be with an optical flat placed in front of the lenses, and a Ronchi grating.

I would like to thank Bart Fried, Dave Groski, and several other people on the Antique Telescope Society website for their advice.

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By the way, these photos show how we held the refractor on the mounting plate for the Ealing mount at Hopewell Observatory.

]]>There is absolutely no information inscribed anywhere inside the cell, inside the tube or outside it, nor on the edges of the lens elements. I can only guess as to what type of glass they used, and figuring it out won’t be easy. The least destructive method I can think of beginning to do this is by weighing them and calculating out their precise volumes, and from that calculating their densities. A graduate gemologist could probably calculate their indices of refraction, but not me.

Tomorrow I plan to measure the curvatures of the lens elements; perhaps someone familiar with old telescopes will then have clues as to who might have made this particular type of optical prescription.

The shims seem to me to be intact, so I think I can rule out astigmatism from lens elements put in crooked. *[OTOH, someone on the Antique Telescopes Facebook group says that the large number of small black spacers in between the lenses may itself be causing the massive astigmatism problem that we found in the star test. I don’t have enough experience to be able to tell whether that’s correct or not.]*

The small chips on the edge of the second (meniscus? Flint?) lens element were already there when I got it. I was also surprised to find that the first (biconvex, crown?) lens element has a small bubble very close to the center. It’s probably not significant, but I will check for strain as well.

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