ch3cooh: (Winter)
Moving to the west coast has definitely worked out well. I currently make a living by teaching for 1-2 different math circle programs every weekday. And the number of circles in the Bay Area is growing: three of the programs I teach for are brand new, and there are plans for more next year. I'm working most extensively with the San Francisco Math Circle (SFMC), which, at present, means teaching for both their middle and elementary programs. SFMC also makes two Math Circles that are, this year, each running their first summer program. The other is Berkeley Math Circle. I'm teaching for both in the month before SPMPS.

The other projects that I'm most excited about right now are:

* Running a public "Carnival of Mathematics" at the end of this Month through SFMC.
* I'm creating some activity content for MSRI/FirstBook. The 5 kids/young-adult books that I'm designing activity materials for are very inspiring. One math history project is themed on "Mathematical Rebels" -- Mathematicians who pushed against the status quo either for a social cause or for a mathematical one.
* I'm making the best 30 sessions from all of my circles this past year into more polished packages that other teachers would be able to use. (In particular, the Rosa Parks afterschool program has asked me to create these and to help train another instructor or two so that the program can expand to more days and sessions than I can take on myself.)
* I've recruited 4 new math circle teachers to take over some of my sessions (with each school's permission). Each new recruit is to design and polish one topic of their own choosing, teaching it for times at four different programs and iteratively developing it on the way. I'm paying these instructors, both for the sessions they take on and additionally for the content-work. And my hope is that this will grow into a larger funnel for pulling more instructors into the math circle community.
* I'm also excited about Proof School's progress. I didn't apply as an instructor, but I do want to connect more with them, which right now means collecting a giant list of math-focused programs in the Bay Area for their website.

Altogether, I'm just about at critical capacity for math circle projects. I've also been making furniture for my and my housemates' apartment. And I made it out to Inyo National Park for two weeks with a friend to study geology over winter break -- that was amazing!

In short, I'm loving living in California so far. I don't think there's anywhere else in the world (yet!) where I could make a full-time living teaching for Math Circles this way.
ch3cooh: (Winter)
I taught a class tonight that didn't work. It was somewhat fun, and somewhat well developed in presenting applications of game theory. But overall: :P I'm writing this post to note down thoughts before I forget, and I'm making it public because I generally think that various try-fail-learn processes should be more open/public. Also, I've been feeling awkward lately that I present myself so unilaterally as an 'experienced teacher' now, and not a student/new and learning teacher or in any other mind-set... more on that in the future perhaps.

I've decided to try for 2 focus goals of self-improvement in teaching each season. This season, my goals are (1) Time management via consciously prioritizing academic and non-academic goals, and (2) creating situations where my students feel ownership for what they're making.

Game Theory part III of IV

This activity was part of a 2 hour session for 45 students in 6th-8th grade.
1) The students cut out paper candy bars and then ranked each type (30 types) of candy bar 0-3 so that the sum of all of the types was 50.
2) In groups of four, I asked them to design an agreed 'fair' method for distributing the candy
3) They implemented the method, and the person to get the most total value of candy (using their own value sheet to scale) in each group, wins a real prize.

Time management went OK. I cut the activity a bit short (~1:15 total out of the 2 horus for the session) and extended some other parts of the class that were working better.

Review: Freedom to Experiment is one of the four foundational freedoms that I strive for when I teach (the 4 are: freedom to fail, experiment, of effort, and to try on different identity roles). But in order for an activity to feel like a valid experiment, there needs to be meaningful choices that the students understand when they are making those choices, and there needs to be feedback, ideally interesting, immediate feedback in response to their choices, and then a chance to go back and make those choices again, more informed. Here's where I think I went wrong:

1) Part 1 mechanics: there were too many types of candy and the max rank wasn't high enough. It should have been something closer to 6 types of candy, 15 points total, rank each from 0-10 I think.
2) Part 2 mechanics: I should have gone over more examples before setting students out to make their own. They had a lot of choice here, and I wanted them to have 'ownership' of their methods, but only a few groups really seemed to construct - most just fished around for something reasonable-seeming and then used that.
3) Part 3 mechanics: OK, but I should have had a sheet for describing more data about the results of using the algorithm -- showing which kinds of preferences or strategies would have the advantage and how and why.

Next time:
I think I should have had a first part of class dedicated to working with 3 or 4 very different division algorithms, letting students get a feel for which are best in different circumstances. Then part 2 could become combining these methods to make something more fair for everyone. Part 3 needed more organization, some follow-up questions with exact answers in addition to the . And I should have left time to play a 2nd time, without letting people change their ranks in-between, to illustrate the impact of making preferences public.

Unexpected:
I spent 10-15 minutes in a few random blocks telling stories about how game theory has come up in scenarios I've seen:

Real estate needing to buy all of some set of houses in order to make any of those houses valuable, a sport conference leader needing to choose a mechanism for coaches drafting their teams, the Splash class lottery algorithm that I worked on as a college student, etc.

These stories hushed the room, and I think I could do more with them. I could ask students how they would 'game' each system, and then how to patch those gaps to make that kind of 'cheat' impossible.

Next time:
1) Real-life game theory examples
2) Trying out 3 or 4 simplified algorithms in groups of 4, with a smaller candy-set
3) Give groups time to build one algorithm from the existing ones
4) Have students implement it
5) Discuss what students chose to do and who it advantaged
6) Play again
7) Discuss the result of more-public value information

Big picture -- If I can make this class into a really excellent one, I think it would make a great introduction to the game theory sequence I've been building. I'm working on 8-12 class sequences right now, of sessions that can stand alone if they need to, but that tie together under a theme as well.

A) Designing Something Fair -- Real Life Game Theory, Types of Fairness
B) Designing Something Fair -- Pet (instead of candy) Intro
C) Designing Something Fair -- Design your own fair division algorithm
D) Designing Something Fair -- Fair Voting Lab

E) Comparing Strategies -- Monty Hall Intro
F) Comparing Strategies -- Prisoner's Dilemma (Iterative Play AI Design)
G) Comparing Strategies -- RPS and Rock Scissors (Mixed strategies)
H) Comparing Strategies -- Real life Game Theory, Nash Equilibria

I) Perfect Play -- Say 16
J) Perfect Play -- 2-pile Nim (poison points)
K) Perfect Play -- Dots and Boxes
L) Perfect Play -- Dots and Boxes AI + Tournament

Higher level Topics:
* Advanced Combinatorial Game Theory
- perfectly logical pirates
- proofs about combinatorial games
- gomoku and variants
ch3cooh: (Ball Pit)
Challenge: write up your initial instinct, then try to prove it and let me know what happens. :D
I really like this puzzle.  And I'm sure that there's a simpler proof than the one I'm using right now.

Puzzle:
GeometryArea

Solution )
ch3cooh: (Fringes of Chaos)
For context, I'm making an activity for high school students in an afterschool math program. The activity is aimed at estimating, understating, and working with really big numbers, and this part of the program (4 sessions) is focused on bijections and infinties, with a theme of how mathematicians think about topics that are too large or abstract to handle with intuition, and for how mathematicians make choices that shape what the mathematics we study looks like.
More Details )

The list so far:

I HAVE NO IDEA, BUT I'D LIKE TO USE THEM IF I CAN FIGURE OUT AN APPROXIMATION:
* The approximate number of primes between 0 and 2^57,885,161 − 1 (???)
* If two 'opponents' at Go are actually collaborating to make the game as long as possible, approximately how many turns might the game last (on a 19x19 board)? (???)
* The approximate number of different chess games for which the board never returns to a previously experienced state. (???)
* The approximate number of atoms in everything living on the earth (???)

I THINK THESE ARE RIGHT:
* The approximate circumference of the earth in miles (~10^4)
* The approximate number of sheets of paper in a stack as tall as the height of the Transamerica Pyramid in SF (~10^6)
* The approximate number of seconds in a human life (~10^9)
* The approximate number of people in the world right now (~10^10)
* The approximate number of neurons in a human brain (~10^11)
* The approximate number of seconds between now and when a dinosaur last lived (~10^15)
* The approximate number of ants in the world right now (~10^16)
* The approximate number of calculations that can be done by the world's current fastest computer (The Tianhe-2 supercomputer) in 1 minute (~10^18)
* The approximate number of stars in the universe (~10^30)
* The approximate number of atoms in the earth and everything on it (~10^50)
* The approximate number of different ways a deck of 52 cards might be shuffled (~10^68)
* The approximate predicted number of atoms in the universe (~10^80)
* The approximate number of rabbits that would exist in the universe after a year if, starting with 2 rabbits, ever day, every pair of rabbits alive mate and produce 6 offspring, and no rabbits ever die. (~10^219)
* The approximate value of the largest known prime number (~10^17425169)

See more at my ask MetaFilter page.
ch3cooh: (Ball Pit)
Being in the right place and having enough time to draw a real picture is hard.  But a few months ago, I found myself really wanting to draw more, and eventually I also found a good compromise: a phone app called "Draw Something." It's effectively digital pictionary, or at least that's the intent.  The best part of the app is that, when you receive a picture, it plays out a 3x speed video of the picture being drawn, instead of just showing you the final image. I like both the drawing and the watching other people draw - trying to guess their word as quickly as possible.  You also get to see your partner watch your drawing and guess your word - an again sped up movie of them putting the letters in place when they realize what the picture is (or sometimes trying and getting it wrong a few times first.)

On top of that, I've been abusing the program in a way - using it to draw entire scenes in addition to communicating the word. :) There's a limit to how much "ink" you can use (which prevents me from spending forever on a single picture).  And there's also an immediate objective and audience every time I draw.  It's been a pretty perfect solution.

Here are some examples, and more of my pictures are hidden below.  And if you think this sounds interesting and want to try it, maybe invite me to play a game with you (my username is "ch12cooh"), or I can invite you if you let me know you're intersted.

app link

Sushi, Mars, Sphinx, Hayride, Ocean...

IMG_20141014_025032IMG_20141012_054738IMG_20141012_063244IMG_20141026_001454IMG_20140521_202826

More pictures! )
And here are a few pictures that other people on Draw Something drew for me:

2014-10-14-00-09-022014-10-18-15-36-432014-10-20-00-20-442014-10-22-01-01-162014-10-24-19-18-52
ch3cooh: (Fringes of Chaos)
These pictures are from the Elementary math circle program at SFSU.  It's a class I teach every Thursday with a coteacher, Josh R. :)  I have never taught with a coteacher this extensively before -- it's awesome!  One of the best parts is that he and I have very different mathematical backgrounds, so whereas the Graph Theory class I'm prepping for next week is foreign content for him, the past 3 weeks that we've spent studying sequences, building up the the Collatz conjecture is new content for me.

The ribbons the students (and instructors) are playing in are sequcens that start with any number of the students' choosing and then progress using the following rules:  If the latest number is even, divide it by 2 to find the next number.  If the latest number is odd, multiply it by three and then add one to find the next number.  The magic is that, so far, every initial choice of a number eventually leads to the same behavior - eventually, the sequence trickles down to the cycle 4-2-1.  But it's unproven if this is true for ALL numbers... it just works for all of the billions tested so far. :D  And sometimes it takes a while for the sequence to get down to that cycle.  For example, our TA chose to start with the number 231 which has a sequence that goes for 127 steps before cycling.


I want to try a similar session out with an older group and combine it with logic tables of pairity. :)  And I think this might be the last part of an "infinity" curriculum that I've been trying to piece together! :D

20141016_180015 20141016_181753 20141016_181547
ch3cooh: (Winter)
Both of these series are amazing.  Mistborn is 3 books long, but part of an enormous unfinished universe.  The three books stand alone though and, if you like epic fantasy at all, or plots that are masterful puzzles in any genera, read these.  Rated on 3 axies: World Building, 9/10, Plot 15/10, Characters 6/10.  The Wheel of Time seems similarly epic, slower, but more intriquete and the characters are a lot stronger in my opinion: World Building 9/10 (but very different), Plot 8/10, Characters 9/10.  The series is 14 books long however, which is... intimidating.

And there aren't any spoilers in the details below, just big-picture writing/style observations. :)

:) I finished the Mistborn series a while ago, but then had a rough and busy couple weeks.  However, wrt the end of Mistborn: IT. WAS. PERFECT!  Really in pretty much all ways I can measure, it was such a satisfying conclusion. The plot and world-building are impossibly well crafted and last quarter of the 3rd book is one ridiculous climax atop the next, paying off absolutely everything. After reading the first book, I thought, "OK, that was good...' but I probably wouldn't have continued the series if not for a very strong recommendation.  Now I pass that recommendation on with equal intensity. :)  And I'm reading the Wheel of Time series now, in part on the premise that if Brandon Sanderson pulled the ending together, it won't disappoint.

However, it's really the differences between the series that interst me/prompted me to write this entry.
More in-depth comparison, still no spoilers though. :- )

ch3cooh: (Ball Pit)
Weighing "Failing Gracefully" highly is one of the best strategies I picked up in college.  It means that among many options for how to do things, giving a strong priority (a "weight") to those that, even if they don't reach fruition, still make a significant positive impact.  Projects that have zero utility until they are completely finished are dangerous - because in reality, few things have the chance to finish before the world changes around them making a 'perfect finish' impossible.  And with a limited amount of time to work on projects, it's also a huge deal when the partial project, put on hold, still has utility in that state.

Last night I ran into a project that has the highest potential to succeed /and/ to fail gracefully that I'ves seen in a long time.  It's not a new idea.  It's simply a local school that wants me to come in once a week for 4 hours to do 4 sequential classes, one with part of each grade.  And the concurrent math classes would be split so that students who need more support with the current school unit have a much higher student to teacher ratio, while students who are ready to move on don't get board/distract the class - instead they come to my session and do something likely related to their teachers' curriculum - but open ended research and problem-solving centric. :)

This role has about 60% of what appealed to me about working at MoMath - outreach to students who might not yet know that math is awesome, non-traditional curricula, visibilty to other teachers and scalability.  And then it also has some remarkable positive differences - that I'd be working with the same group of students, building a community and relationships with them over an entire year for instance.  Also, more freedom in what and how to teach.  And it's /so/ tempting that the scalability is still there... just doing this for 4-5 schools myself is a lifelong career that I think I'd be pretty happy with, in part because of how much free time it would leave me with to do any projects I saw fit to do.  :)  But, the large scale dream would be collecting people to do this with me - maybe from the math-circle teaching community.  The MC lesson plans are a good fit, and what I'm doing now -- developing MC-style lesson plans that can parellel core 'school-content' could be useful to other people.  I can see life that I would really like in this -- teaching 2-3 days/week, and coordinating the administration of a dozen people doing similarly, observing their classes 2-3 days/week.  Constantly building and rebuilding lessons to fit what new schools and teachers are doing. Working with teachers to expand their regular curriculum with the support of these weekly sessions.  :D  Wow -- it'a a position and a lifestyle that meets almost every metric I've got.

So yeah, I'm excited.  And... I think there's only a 5, maybe 10 percent chance that what I'm describing above actually happens.  But the failure is doing this myself at one school and then it doesn't go further than that.  And that 'failure' is also OK - very OK.  A lot of what I'm doing right now is like this -- I'm not sure what part will 'take off' if any, but the partial steps -- teaching for a lot of Math Circles and trying out other types of teaching/tutoring as well - the partial steps are also good things.  :)
ch3cooh: (cake)
I am booked for a conspicuous number of one-way flights over the next few months and a LOT of travel in general.  Here's a graph :) (dates are approximate).

flightgraphs

And Some Awesome Reasons
1) Buying a car in NC
2) Moving 'home-base' to CA and visiting many awesome people and places on the way
3) Visiting Rafael's Family in CA
4) Going to the beach with my family in NC
5) Visiting MathPath in MA
6) Teaching for SPMPS in Upstate NY
7) Returning to CA for the foreseeable future
ch3cooh: (Balance)
Hi LJ!  Big news:  I'm switching coasts.  There are a lot of reasons why, and there is especially one reason: a gentleman who has been sending me puzzles for almost a year now (among other ways of keeping in touch. ;-)  His name is Rafael, and he is a current student at Stanford, so moving to be only an hour's drive away from him (Berkeley) instead of 2500 miles away is... exciting. :)

But I'm still wrapping up here in NY/NJ for the next few months and I could definitely use a little fun distraction during that time.  Also, I haven't organized a crazy experiment/game in a long while! So....

QUOTES ILLUSTRATED

This game is motivated by wanting to

  1. Know which pieces of books most stand out to my friends.

  2. Do art -- watercolor in particular -- I just acquired a watercolor kit and I want to play with it. :)

If you have a quote, please add it in the comments below (with which # you'd like it placed at 1-15) and I will illustrate it -- I'm going to try to do three illustrations every week in April and Early May. But I don't want to steal all the illustrating fun, so if my intended role in this 'game' sounds like fun to you, feel free to steal it and add your own illustrations here: http://entropyinprogress.wikispaces.com/Quotes+Illustrated --- ultimately, this site is where all of the quotes and images will end up in addition to irregular LJ entrees over the next month.5

An Example:

"Marco finds her shawl left behind in the game room, still draped over his jacket." (quote context)
-- Erin Morgenstern, The Night Circus (pg. 222)
20140405_122131     

Quote Scheduling )
ch3cooh: (Cell Phones)
Woot! I survived the week!
MoMath has been very very full of school groups for the past few weeks.  I've been teaching 5 or 6 classes nearly back-to-back every day.  Mainly Cryptography though, which is my favorite of our three classes.
But I'm finally settling into the 'routine' of this, and re-finding the time and energy to do other awesome projects at home on the weekends. :)

Way back in January, a bunch of friends sent me amazing pictures... and I got hosed and hoarded them all to myself.  No more of this.  Here are the results (my favorite pictures) from week 2 (of 4)  More soon.

Category 1:  ID -- Can you ID the subject matter of each of these shots?  (hints behind the cut)
Capture(Hidden)
hints/full images )

Category 2: Sometimes the subject matter is simply awesome
Capture(Things)
full images )

Category 3: Beautiful Locations
Capture(Places)
full images )
ch3cooh: (Yosemite)
In short: a month ago today, the New Museum of Mathematics opened in Manhattan, NY. 
Soon after that, I received an email with this video of the MoMath Opening
Soon after that, I applied... then an interview :) ... then a job offer :D... and, a couple days ago, I accepted the offer! :D!!!
I'm going to be an Education Coordinator at the museum starting in February!
I'm planning to move to an apt off the PATH line in New Jersey, since PATH drops right into Manhattan at Madison Sq. Park where the museum is located.
So if you want to visit me and/or the new museum sometime after March 1st, just email me regarding 'crash space'.  I <3 NY, but It's more than a little terrifying to be uprooting and relocating so suddenly! I am definitely going to miss friends in Boston, and I'd love to have you all visit! 
Momath

More About the Museum (and PICTURES!) )
ch3cooh: (Yosemite)
Here are the results of the 'first week'  There were more pictures than I expected, so I've decided to post 15 of those pictures this week, in 3 categories of appreciation.  A lot of people put more effort into framing shots than I anticipated would be a part of this game.  So this week's categories are all about framing: choosing the time, location, and composition of the shot purposefully and carefully. :)

The categories for next week will be: "portraits," "the unseen," and "energy."

For this week, the categories (retrospectively declared) are:

PerspectiveFull Images )
Perspective

TimingFull Images )
Timing

LightFull Images )
Lights

I think I can officially declare this game a success!  It's been great to get pictures from a dozen or so people every day, and single shots from a few more friends.  Thank you! - to everyone who sent in anything! :) 

Read more...A few more pictures that I took from my NC->NY airplane! )
ch3cooh: (Cell Phones)
An old friend of mine from elementary/middle school sent me the very first image of the year:
KatieD1-1-12
Isn't it beautiful!  :D

GardenEDU

Dec. 12th, 2012 12:12 pm
ch3cooh: (Yosemite)

Interested in mathematical origami or Swedish folk music?  How about online education or filmmaking?  If so, you might want to take a look at the project I'm working on, a site at gardenedu.org.  It's a page of video-lessons organized in 2 trees of subject matters: "Swedish Folk Music" and "Origami Design."

aesthetics0

First of all, I hope you just think it's pretty.  I had a lot of fun doing the art - and the visual metaphore is that 'topics' are trees suspending glowing spheres: 'concepts' so to speak.  I'm not enough of an artist to fully demonstrate how beautiful this idea looks to me:  a forrest, (perhaps more like a garden for the variety ;-) full of softly glowing spheres connected by strings of smaller lights, like a neural network of lights draped over a living forrest.  If I could, I'd make a video in which you zoom out now and see that this lit forrest is itself inside a glowing sphere with thin lines connecting it to nearby concepts in aesthetics, education, and engineering:

So, in that order:

AESTHETICS )EDUCATION )ENGINEERING )

And more feedback/recomendations/insights on what's up already would, of
course, be appreciated!  So comment or email me if you have ideas! :)

ch3cooh: (light on black water)
I'm about to teach a summer math course: Proofs and Visualizations (it's gonna be awesome!) and I've been talking about the problems/units with a bunch of friends.  One of the units is based on a math puzzle called "The Paradox of Perfectly Logical Pirates"  I've outlined the problem below.  However, here's the glitch - once you've come up with the straight-forward answer, here's another question:  Two twin brothers, Alfie and Ben are given $10 of allowance every week.  More specifically, Alfie's given the money and his father tells him to split it between himself and his brother Ben.  As long as Ben accepts the split, the boys get to keep the allowance, but if Ben ever complains, neither boy will get anything.  Alfie and Ben are logical kids, and both realize that something is better than nothing.  So, each week, Alfie gives Ben $1 and keeps $9 for himself.  And Ben doesn't complain because $1 is better than nothing.  But then, one day, Ben goes over to his friend David's house.  David's twin sister, Kate, has just been given $10 allowance to split between herself and David - their father has the exact same rules.  Perhaps David is more sure of himself or something, because he's decided that he'll reject the split unless his sister Kate gives him $5.  Since Kate wants to keep getting allowance ($5 is better than nothing) - she splits the money evenly between them each week.  Ben sees this and is surprised and impressed.  Can he get the same results from his brother Alfie?  Later that week, Ben visits his friend Fanny's house, although Ben doesn't like Fanny all that much - she's  a bit greedy and reminds Ben of his brother Alfie.  When Fanny's brother Elmo gets $10 of allowance to split between the two of them, Fanny tells Elmo that she'll complain to father unless Elmo gives her $9.  Elmo believes her (she is a greedy one) and would rather get $1 than nothing, so each week he gives Fanny $9.  Sometimes Ben wonders if he could get Alfie to do this for a while at least - just in order to pay him back for the many weeks of taking $9 for himself. 

In real life, who has the power? - the person splitting the money, or the most stubborn and greedy player? Why?  How can this question be described rigorously mathematically? 

The Paradox
of Perfectly Logical Pirates )

I'm really thrown by these puzzles/'paradoxes' and I feel like there's got to be a good theory somewhere to explain what's going on.  I also think these questions are practically relevant since they suggest that things like mutinies might have a logical basis, whereas simple economic theory frequently predicts that the masses will just accept their short-changed fate. 
ch3cooh: (Engineering Hubris)
The Problem:

Solution:
Hot Chocolate!

Just drink it at a nice, steady pace. :)
People make life so complicated!
Thankfully, hot chocolate additionally makes life delicious. ;-)

HAPPY HOLIDAYS! :D
ch3cooh: (Default)
LJ! Hi! you haven't read all of the same material, but I think these ideas are awesome, so, over the next few weeks, I'm going to post about them and see if I can maybe even have some awesome discussion.

The realistic and honestly pretty terrifying version of parallel universes that split exponentially as the realization of every choice that could have been made

Awesome things related to this topic:
This story by Borges: The Garden of Forking Paths
This painting by Escher: Angels and Demons
This startrek episode, particularly after 35:00: Star Trek TNG Season 7, Episode 11: Parallels
This old batman/superman movie: Justice League: Crisis on two Earths
and a little bit of Lovelock by Orson Scott Card and even some Fight Club (an awesome movie that I swear I won't spoil at all)

I cut this massive wall of text for your convenience, but if you read and/or respond to it... <3 <3 <3 )


ch3cooh: (Topspin)
I'm so sorry I forgot to take a picture on my cell phone, it only lasted a minute!  I was walking down an easement  (the passes of rough gass and shrubs under telephone lines or above pipes that require occasional vehicular access) at midday and it was very sunny, but there were a few thuderclouds threatening, though not overwhelming the sky.  They burst out in a warm one-minute downpour, and a girl, a kid who was playing in her back yard nearby shouted to me, "look, a rainbow!"  I looked in her direction and, between us, there was, not one but a series of increasingly clear partial arc rainbows - the one furthest to the right being the most shockingly bright rainbow I'd ever seen.  The mist creating them was thick since the warm leaves kept the moisture of the downpour in the air longer.  I observed the rainbow for a moment, but then kept walking down the easement.  It was still raining in scattered downpours, thundering even, and there was a flash directly above me, but the lightning remained above and what came down instead was a new ray of rainbow that shot down and anchored in the ground beneath me, passing through me, and I could feel it, though it was not solid like fabric, much more like light warms you or a magnetic field might raise the hairs on your arms.  It was not solid, but it was as vibrant as the most vibrant rainbow before, and more ribbons shot down all around in the easement where light penetrated to the ground, and all of these rays anchored in the ground - maybe a dozen altogether.  They were thin, no more than five inches across the span of the colors and did not seem to stop just above or even at the ground, though I knew they could not have really penetrated the soil in any way.   Still, I was surprised that I could see where they met the ground as most rainbows I've seen have had edges that fade out, gradually, and I felt like these would be the kind inspiring people to believe their might be gold in the ground beneath the anchors.  But, what was most beautiful... When a rainbow arc would have hit a tree on the fringe of the easement instead of proceeding unobstructed to the ground, instead it wound about the tree in a tighter and tighter spiral containing the whole tree, tightening in diameter like a cone around the base of the tree where it constricted around the tree in a semi transparent, but incredibly bright ring, the stripes of the colors parallel to the ground.  The kid had run into the easement and up to one of the brighter spiraling rainbows, she saw that I looked confused and commented that "they have to spiral since they cannot have an end, but since the space is finite, they must coil infinitely at the base."  Which made sense in an odd way, like a particle in a box...  But the downpours had stopped and the rainbows were fading.  I wished that I had a camera, and then realized that my cell phone would do, but by the time I got it out of my pocket and into camera mode, nothing remained bright enough for my cell-phone resolution to capture.  So, no picture! sorry!

also, it was a dream... so I'm not sure how well the film would have developed.  But, god it was beautiful! 
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