Welcome to ZenFox42's Web Page

As you'll see, I love to learn new things. I also have a bad memory, so I write things down so I can look at them later. What follows are some of the things that I have researched over my life, in no particular order...


I've been taking pictures since I was a kid (thanks, Dad!).
Here are just a few of my best photos, all taken on slide film and then scanned.
The only digital manipulations have been for color balance and brightness.
            There's less than a dozen, take a look! At the very least, check out the "red wagon" and Monarch butterfly photos, they were once-in-a-lifetime shots!

General Relativity

I spent several years learning as much as I could about general relativity. The following files represent a summary of general relativity, broken into bite-size pieces of roughly 10 pages each (except GR2f, which is more like 20). In all these files, items in blue text are the most important things, items in green text are "gotchas" for the beginning student of general relativity (except for in GR1f), and items in gold are things I am 99% (but not 100%) sure of.

DISCLAIMER #1 : these were originally created as my own private notes, so sometimes they explain things in detail, sometimes they just present the results. They are not "lessons" or "tutorials". Some of the labels in the figures don't always exactly match the text, but I tried to indicate that where possible.

The first set of files contain the minimum amount of information you need in order to understand what Einstein's "field equation" means. They assume the reader is already familiar with college-level math including vectors, derivatives, partial derivatives, and matrix math, as well as introductory physics. If you want to skip the math, the end of GR1c contains the explanations and GR1e contains many cool visuals. There are references in these files to the GR2 files, but you do not need to read the second set in order to understand these.

GR1a – a brief review of vectors and coordinate systems; an introduction to tensors, index notation, and Einstein

summation; definitions of manifolds, tangent vectors and tangent planes

             GR1b – a brief refresher of special relativity; basic terms and concepts of general relativity; the metric tensor; an

introduction to covariant and contravariant tensors; introduction to parallel transport

             GR1c – the Riemann, Ricci, and Weyl tensors; the Ricci scalar; an introduction to the stress-energy tensor or

energy-momentum tensor (it goes by both names); Einstein’s general relativity equation and some

explanations/interpretations of it

             GR1d – tensor notation is incredibly “dense”, so this shows Einstein’s equation (in terms of the metric) expanded

into its full glory

             GR1e – some visualizations of common spacetimes including Schwarzschild (non-rotating mass), Kerr (rotating

mass) and a moving mass (the visualizations of the Kerr spacetime are my own, and try to show how spacetime is stretched around a spinning mass); other spacetimes such as

wormholes and warp-drives; a discussion of the many tests general relativity correctly predicts

             GR1f – general relativity as applied to cosmology, and why we think the universe is filled with “dark matter” and “dark

energy” (and all the assumptions that go into that)


These next files go into much more mathematical depth and detail, explaining things well enough so that you could (hopefully) understand papers and articles about general relativity.

             GR2a – more on vectors and tensors (wedge product, dyadic product, Levi-Civita symbol, etc.); more general

relativity terms and definitions; intrinsic/internal vs. extrinsic/external curvature; various kinds of “flat”


             GR2b – much more on special relativity : spacetime diagrams; simultaneity and cause/effect; light cones; Minkowski

diagrams; Lorentz transformations; doing physics in special relativity (velocity, acceleration, momentum, force, etc.)

             GR2c – all about derivatives : path tangent vectors; scalar field derivatives; directional scalar derivatives; vector field

derivatives; affine connections; Christoffel symbols; covariant derivatives; directional covariant derivatives; Lie

derivatives; the explanations of the Christoffel symbols and covariant derivatives contain

as-clear-as-I-can-make-them explanations with good visuals

             GR2d – all about covariant and contravariant tensors : how they are defined and how they transform; the Jacobian;

one-forms; exterior derivatives; and a recap of contravariant/covariant formulas

             GR2e – electromagnetism : classical formulas; special relativity formulas; general relativity formulas

             GR2f – general relativity, take two : differences from special relativity; affine parameters; proper acceleration; a recap of

parallel transport and geodesics; an in-depth as-clear-as-I-can-make-it look at the stress-energy-momentum tensor

(including my own standard-physics explanation of why “pressure” draws matter and energy towards it); energy

conditions; more on the Weyl tensor; geodesic deviation; a discussion of the pitfalls when converting line elements

to metrics; the meaning of off-diagonal terms in the metric (which is rarely discussed anywhere!)

And regarding the off-diagonal terms in the metric, I found that there is a region around objects where the angle between the axes of space-time become complex (imaginary),
which is completely a different region than the ergosphere! Click here for a description of that surface.


I originally had in the back of my mind an idea for a third set of documents which would include enough information so that you could actually work problems in general relativity, but that probably won’t happen at this point.  Some of the GR2 files reference these non-existent GR3 files.


DISCLAIMER #2 : I shamelessly copied text and figures from other people’s web sites to create these notes, but did not keep track of what came from where.   Some of it is copied verbatim, much of it has been edited, rearranged and merged, and some of it is entirely my own.  If you find any of your own material here that you would like me to reference, please let me know, I would be glad to do so (I didn’t keep track of what came from where initially).  If you would like me to remove your material, I will, but please realize that if I used it that means that I thought it was exceptionally clear and helpful. 


As I continued to look into general relativity, I discovered an amazing thing : Einstein's field equation is a simplification! The equation, as complicated as it is, ignores something called "torsion" (which is what happens when you turn a screwdriver or wrench). Including torsion adds a second, equally-as-complicated, equation to the mix!

On a related note, I also studied Quantum Mechanics (QM) for a while, and it describes the world at the atomic level, where many many weird, counter-intuitive things happen.
But here's what I found to be the oddest thing about QM : it's the only branch of physics where the equations work perfectly (match experimental results), but what they mean is still open to debate over 100 years after their discovery!
By that I mean that everybody agrees with what F=ma or even E=mc2 are saying about the physical world. Not so with QM!
Click here for a description of many of the different ways QM can be interpreted
Citizen's Rights

After watching way too much "Law&Order", I began to wonder how much of what the police were doing in the show was technically legal. I also wanted to know what my rights were when dealing with the police (not that I ever expect to need it, I was just curious). Nowhere on the web could I find a single source for all that, so...
Click here for a discussion of your Constitutional rights as a U.S. citizen when you interact with the police
       A few teasers : did you know that -
            Nowhere in the Constitution does it guarantee a "trial by a jury of your peers" or that you are "innocent until proven guilty"
            Even tho the Constitution guarantees your right to a trial, the vast majority of convictions do not involve a trial
            Even tho the Consitution says you cannot be arrested without a warrant, the vast majority of arrests are legally made without warrants
            Even tho the Consitution says you and your home cannot be searched without a warrant, the vast majority of searches are legally made without warrants
            Also, anything you say to the police at any time may be used against you, even if you haven't been "read your rights"!
       This has been reviewed by an ex-policeman for accuracy!

Multiple Universes

Ok, we all know that infinity is BIG. Really, really big. But if the universe is truly infinite (which current measurements support), then it is guaranteed that all of these exist somewhere in other galaxies far far far far far far far far far away from here :
            There is another planet that looks just like Earth, with humans who have our exact same history, except that the Germans won WWII
            There is another planet that looks just like Earth, with humans who have our exact same history, except that we lost the Revolutionary war
            There is another planet that looks just like Earth, with humans who have our exact same history, except that Al Gore is (or was) President
            There is another you sitting on an identical Earth reading this web page
            There is another you sitting on an identical Earth reading this web page wearing a different shirt
            There is another you sitting on an identical Earth reading this web page living in a different house
            There is another you sitting on an identical Earth reading this web page who knows Elvis personally (and who's still alive)
            There is a guy named Sherlock Holmes whose life is exactly like everything we've ever read about him
            There are people who experienced everything described in "War and Peace", "Gone With the Wind", "Pride and Predjudice", etc.
            There's a creature reading a fictional story about the exploits of an alien named Ghandi that is exactly like his life here
            (Sorry, sci-fi fans, no Star Wars or Star Treks out there unless warp drives and light sabres are actually physically possible)
Click here for a mind-blowing discussion of the various kinds of infinite multiverses (multiple universes) we might be living in
            This is based on real science!

Robin Hood

Everyone knows about Robin Hood, but the earliest stories portray him completely differently than the swashbuckling Errol Flynn or the accent-less Kevin Costner. His legend started as early as the 1300's, and we have many complete ballads about him from the 1400's (!). In those, he has nothing to do with King Richard, Prince John or Maid Marian, rarely fights injustice for a greater good, and never steals from the rich to give to the poor. These elements weren't added to his legend until around 1500.
Click here for a discussion of all the earliest available materials on Robin Hood

The Ancient Celts

After watching a group of people who called themselves Druids doing a ritual at Stonehenge on the Discovery channel, I began to wonder out of idle curiosity just how much of what they were doing was based on any kind of factual evidence. Being a graduate student at the time, and prone to do research at the drop of a hat, I began to look things up in the library.
Click here for a detailed discussion of the ancient Celts, their culture, religion, and of course the Druids, based on current [2001] academic understanding
       I read two dozen books and summarized them, so you won't have to!
       BTW, I grew up with "BC" and "AD", so that's what I use. If that offends your sensibilities,
             please convert them in your head as you read (I also call the dinosaur with the really long neck
             a brontosaurus, and Pluto a planet)

After reading so many books that said "Strabo said this" or "Caesar said that", and then seeing so much outright misinformation on the web about who said what, I finally decided I wanted to find out for myself.
So I spent hundreds of hours in the library scouring the ancient translations, finding references to the Celts, and collecting them together.
Click here for a summary of almost EVERYTHING the ancient Greeks and Romans said about the ancient Celts.
Copyright issues prevented me from including the actual material, but even with just the summaries, it is over 80 pages of what 100 ancient authors said about the Celts.

Ancient Chinese Philosophical Documents

Click here to read my own translation of the ancient Chinese Zen classic, 信心铭
Otherwise known as "Hsin Hsin Ming", "Xin Xin Ming", or "XinXin Ming", which is usually translated as "Faith Mind" or "Faith in Mind"
This is an early Zen Buddhist document that shows its Daoist origins
(There are a couple of other, earlier versions of this translation "out there" on the web; this is the most recent one.)
       Click here for a behind-the-scenes look at my translation process

Click here to read my own translation of the ancient Chinese Daoist classic, 內業
"Nei Ye", "NeiYe", "Nei Yeh", or "Inner Cultivation", "Inner Enterprise", "Inner Training", "Inner Development"
       This earliest known Daoist document pre-dates the Dao De Jing!
       Click here for a more in-depth look at my translation process

Click here to read my own translation of the ancient Chinese scripture, 清靜經
"Scripture of Clarity and Tranquility", or "Purity and Stillness", etc.
       This document has elements of both Daoism (it quotes the Dao De Jing!) and Buddhism.
       Click here for a more in-depth look at my translation process

After translating the above short documents, I finally decided to try to translate the Dao De Jing, or Tao Te Ching in the old spelling.
This translation attempts to be as literal as possible, and shows you what English word each Chinese symbol stands for.
Click here to read my own translation of the Dao De Jing

While I was translating the Dao De Jing, I ran across the idea that maybe it had been "compiled" over perhaps a century or so from several different sources.
Click here for an analysis of the "voices" or "layers" in the Dao De Jing.

Pattern Recognition

When I discovered Pattern Recognition (PR) as a grad student, I fell in love with it. It's something that animals and humans do very well, but has been very hard to get computers to do. PR is what we use every day to recognize spoken words, our friend's faces, and the letters on this page. Animals use it to differentiate between their own kind, predators, and food (without it, they wouldn't run from what wants to eat them, or they'd try to mate with their food - either way, they wouldn't last very long). In our homes, dogs and cats use PR to recognize their name, and the word "no" (altho cats usually choose to ignore the latter). I was so interested in PR that I decided to explore it for my PhD thesis. I applied PR to analyze the data from an electronic nose (e-nose). Yes, there really is such a thing as an electronic nose! It consists of multiple sensors, each of which reacts slightly differently to minute traces of chemicals in the air. PR can differentiate between the sets of sensor reactions, thus identifying the different odors. The following links are some of the conclusions and discoveries I made during my PhD research.

Click here to find out the best estimator to use for pattern recognition problems with small samples

Click here for a discussion of why you CANNOT use PCA or LDA graphs to determine how well a multi-class dataset is separated

Click here for a description of a new way to project multi-dimensional data onto a two-dimensional graph

Click here for an explanation of why correlation between features should NOT be used in feature selection


I like to cook, and have way more recipes than these, but these are the most unique.
Real, no-cleanup hot chocolate!
Fruit Ices
Thai Chicken and rice - all my favorite Thai flavors!

Savage Worlds

I've been playing role-playing games for decades, and finally found a system that I really like called Savage Worlds. In order to give back something to the SW community, I came up with these "summaries" which I hope explain the rules a little more clearly, collected together in one place. These are all rules-as-written, with clarifications from one of the game developers.

Here you will find a complete list of combat actions, as well as how to handle Cover, Healing, Using a Power, Grappling, and Chases.
Also ideas on how to handle Persuasion, make longer-lasting Bosses, a better Combat Rating, and a generic way to break up any science-fiction setting into various "future tech levels".
And a system-independent description of various "tech levels", "technology levels", or "technology ratings" over the ages.

My Belief System

I've spent several decades thinking about how I think the greater universe works. I've drawn on many sources (too many to list anymore), and followed what "makes sense" or "feels right" to me.
Here you will find a very brief description of what I believe about god, the soul, reincarnation, karma, etc.

I've also written down thoughts that came to me as I was reading, and collected them here.
Here you will find these "Reminders to Myself".

A quartz "chi" detector

Ok, I'm pretty much a "hard-science" guy - I don't believe Einstein was wrong, and everything I presented above in General Relativity is by-the-book.
But, I also know from first-hand experiences that there is more going on in the universe than can be explained by science. And everyone's allowed *one* way-out-there idea, right?

Click here for a detailed outline of how to build a device that might be able to measure the life-force the Chinese call "chi".