### Virginia Trimble on Pyramids and Missing Extraterrestrials

Did you ever wonder if there was any 'real science' behind the subject matter of shows like 'Ancient Aliens'?

There is! Yesterday I came across these two articles[1][2] by Dr. Virginia Trimble: UC Irvine astronomy professor and science historian. The first article was written by Dr. Trimble when she was still an undergrad at UCLA. It detailed her calculations showing that the constellation Orion was visible through the 'so called air shaft' of Cheops pyramid when it was constructed.

The second article was an excellently written and fun to read review of the a 1979 "Symposium on the Implications of Our Failure to Observe Extraterrestrials " The symposium was attended by luminaries such as Freeman Dyson and Bracewell. Among other topics, there were discussions of how many planets were available that could support life. It gives a nice historical perspective on today's post from John Baez [3]

References

1. "Astronomical Investigation Concerning the so-called Air-Shafts of Cheops's Pyramid"

2. "Where Are They? A Report on a Symposium on the Implications of Our Failure to Observe Extraterrestrials "
http://www.bigear.org/CSMO/PDF/CS06/cs06p20.pdf

3. John Baez on Earth-like planets near Red Dwarfs

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

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The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differe…

### Division: Distributing the Work

Our unschooling math comes in bits and pieces.  The oldest kid here, seven year-old No. 1 loves math problems, so math moves along pretty fast for her.  Here’s how she arrived at the distributive property recently.  Tldr; it came about only because she needed it.
“Give me a math problem!” No. 1 asked Mom-person.

“OK, what’s 18 divided by 2?  But, you’re going to have to do it as you walk.  You and Dad need to head out.”

And so, No. 1 and I found ourselves headed out on our mini-adventure with a new math problem to discuss.

One looked at the ceiling of the library lost in thought as we walked.  She glanced down at her fingers for a moment.  “Is it six?”

“I don’t know, let’s see,” I hedged.  “What’s two times six?  Is it eighteen?”

One looked at me hopefully heading back into her mental math.

I needed to visit the restroom before we left, so I hurried her calculation along.  “What’s two times five?”

I got a grin, and another look indicating she was thinking about that one.

I flashed eac…