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Monthly Archives: October 2021

A Navigation / Geometry Problem

24 Sunday Oct 2021

Posted by gfbrandenburg in astronomy, History, Math, science, teaching

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Benjamin Banneker, District of Columbia, Pierre l"Enfant, Washington

I had the pleasure of helping lead a field trip for 9th grade Geometry students at School Without Walls SHS that we call ‘Math on the Mall’ assisting with two colleagues from the SWW math faculty.

One of our goals is for the students to see how beautifully and geometrically this city was laid out by Pierre l’Enfant, Andrew Ellicott, and Benjamin Banneker about 230 years ago.

While there are lots of myths written and repeated about Banneker’s actual contribution, the fact is that he was the astronomer, who was responsible for determining due north, exactly, and the exact latitude and longitude of the southern tip of the original 10-mile-square piece of land. With no Internet or SatNav or even a telegraph or steam engine, but with a very nice refractor and highly accurate clock that he was entrusted with, but with no landmarks to measure from, he was able to do so, in 1790.

I was sad to see that exactly none of the students know which way was north – in a city where the numbered streets near the Mall and the rest of DC’s historic downtown were almost all laid out perfectly north-south, and the streets whose names begin with letters or words like ‘Newark’, and the streets along the Mall, are all laid out perfectly east-west. Very few of them had ever seen the Milky Way, though most had heard of Polaris or the North star.

Hopefully they will remember that in the future as they do more navigation on their own in this great city.

I challenged them to try to figure out why the angle of elevation of the North Star is the same as their latitude. Here is a diagram illustrating the problem:

The Earth, Polaris, and You.

This diagram is intended to help you understand why the North Star’s elevation above your horizon always gives you your  latitude (if you live north of the Equator.

The big circle represents the Earth. The center of the earth is at E. The equator is AD.

YOU, the observer, are standing outside on a clear night. You see Polaris in the direction of ray BG. Line HCE is the Earth’s axis, and it also points at Polaris – which is so far away, and seems so tiny, but yet is also so large, that yes, parallel rays BG and CH do, for all practical purposes, point at the same point in the sky. Ray ED starts at the center of the Earth, passes through you at B, and goes on to the zenith (the part of the sky that is directly overhead). The horizon (BF) and the zenith (ray EB) are perpendicular. Also, line HCE (the earth’s axis) is perpendicular to its equator (segment AED).

Using some sort of angle measuring device, if you are out on the National Mall at night, you can very carefully measure the angle of elevation of the North Star above the local horizon, and you should ideally find that angle, FBG, is about 38.9 degrees, but we could also call it X degrees.

Prove (i.e. explain) why your latitude (which is angle AEB) measures the same as angle FBG.

What are the givens?

=========================================================

Full disclosure: My daughter graduated from SWW two decades ago, and I taught there as well for a year and for 10 years at a school that is now associated with it: Francis (then JHS now a middle school).

The kids were nice back then, and they still are. I thought the teachers did a great job.

This is a DC public high school that you have to apply to.

Benjamin Banneker was an amazing person. There are a lot of myths that have been attached to his work and accomplishments, which I am guessing might be because those people didn’t actually understand the math and astronomy that he did accomplish. The best book on him is by Silvio Bedini.

‘Math on the Mall’ was originated by Florence Fasanelli, Richard Thorington, and V. Frederick Rickey around 1990. I participated as a math teacher in a couple of those tours led by FF. Later, I wanted to take my students on a similar tour that would include a trip to see a number of the works of the geometer and artist Maurice C. Escher, and couldn’t find my copy of their work, so I made up my own, and added to it using the work of FF, RT, and VFR and suggestions from teachers and students. Later on, the Mathematical Association of America made something similar, which you can find here.

My version was on the website of the Carnegie Institution for Science for a number of years. See page 56 on this link. I need to find someone to cut out some of my excess verbiage and then trot it out to a publisher.

Correct weight of the mystery glass

13 Wednesday Oct 2021

Posted by gfbrandenburg in Hopewell Observatorry, optical flat, Optics, Safety, Telescope Making

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mystery glass

More progress with the 22-inch wide, 4-inch thick mystery glass.

It took four of us old farts- Jim Kaiser, Alan Tarica, Tom Crone, and me – to extract the mirror from its case (which was located under a very heavy Draper-style grinding and polishing machine) roll it onto a little stand we fabricated on a decent gym scale I borrowed from my gym ( http://www.True180.fitness ) and weigh it.

We were very careful when moving that heavy mirror. Nobody got hurt in any way. When putting the mirror back into its sturdy wood carrying box, we used ancient Egyptian technology of little rollers, and it worked like a charm.

The bathroom scale we had used earlier, up at Hopewell was very, very wrong. We found that the weight of the glass was really 212 pounds (about 96 kilograms, or 96,000 grams), not 130 pounds. Its volume was 20,722 cc, so its density is roughly 4.6. Will have to see what types of glass have roughly that density and an index of refraction of about 1.72 to 1.76.

I heard from one veteran telescope maker:

“I’ve been in the Tucson astronomy club for many decades and also in the optics industry there. Most all institutions that had connections to astronomy or optics in the 60s got portions of several semi loads of “glass bank glass”, glasses that at one point in the past were considered strategic materials for certain optical designs/systems. There was a wide variety of materials, but almost all was identified in some way. We’re there any markings ar data scribed in the glass? The largest I saw was about 15”, so yours might be a different source.

“A co-worker of mine has identified several mystery glasses from an accurate determination of density. Seems like you should be able to get better results w/a more accurate scale. Also many glass types made decades ago are obsolete – my friend has some older glass catalogs that might help you determine what it might be with more accurate numbers.”

So these were cast-offs from the Military Industrial Complex, basically: pieces of glass that the military decided it no longer needed for projects that had either been completed or abandoned, and that they didn’t feel like storing any more. So they gave them away to groups like National Capital Astronomers and Hopewell Observatory.

The only markings on the glass are the following: a heavily inscribed (by hand) apparent date of 2-8-56, which probably means either February 8 of 1956 or the 2nd of August 1956. Judging by the handwriting style of the numerals, it was probably Feb. 8 of 1956 (US style). Under that are the numerals 0225, which we have no idea about. In pencil, someone with US-style handwriting wrote what looks like “Low #” in cursive. Again, we have no idea what that means.

Thanks so much, Jim, Alan, and Tom!

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