The Fibonacci Sequence and Golden Ratio

I just spent the last 3 hours figuring out why this is cool. I’ve never considered spirals so intensely in my entire life. You should have been there; it was spectacular, especially when my head exploded. By the way, I was doing these derivations and proofs on paper at 2AM so it could all be entirely inaccurate.

The ratios of the numbers in the Fibonacci Sequence oscillate around and converge on the Golden Ratio, phi=1.61803399, which can be derived by solving phi=a/b or phi=(a+b)/a, where a=(b/2)(1+sqrt(5)) (which can be demonstrated geometrically quite nicely apparently [1:25-1:40]), which yields phi=(1+sqrt(5))/2).

The Golden Ratio does not describe spirals, but specifically the spiral where roughly the effective diameter of the last 180* becomes the effective radius of the next 90*.

I suspect that The Fibonacci Sequence and The Golden Ratio in relation to “spirals occurring in nature” are applied too generously and abused across the internet. In any case, the strange obsession with The Golden Ratio has never sat well with me so I spent several hours mapping out my arguments against it, but then in attempting to prove my own theories I kept proving why the Golden Ratio is awesome. I only have a few more issues and I was half way through trying to determine the angular frequency (or range) pattern which sunflower seeds (or pine cone leaves or pineapple spirals) must follow to give over-excited Fibonnacci and Golden Ratio fans the ability to claim complementary spirals both occurring at rates that correspond to numbers in the Fibonnacci sequence but that’s when my head exploded.

What is the scientific name for a pine cone leaf?

Have fun, but more importantly, be happy.

This may be obvious, but sometimes it’s easy to confuse the two, especially while in the middle of a fantastically fun mess, but fun is certainly not happiness and I’d like to take special care to remain aware of this difference over the next few weeks. I believe happiness is far more important than having fun or being fun. Fun can be a positive characteristic or a good time, but happiness is a fulfilled state of being, fun or not. I could live well without fun, but would be miserable without happiness. Moving forward, I will focus on developing happiness in myself and my environment and will welcome fun only as it serves this greater purpose. Have fun, but more importantly, be happy.

The Art Of Learning: A Journey in the Pursuit of Excellence

The Art Of Learning: A Journey in the Pursuit of Excellence
Josh Waitzkin

“…to be quietly, intensely focused, apparently relaxed with a serene look on your face, but inside all the mental juices are churning. You flow with whatever comes, integrating every ripple of life into your creative moment. This Soft Zone is resilient, like a flexible blade of grass that can move with and survive hurricane-force winds.”

“When injured … I try to avoid painkillers and to change the sensation of pain into a feeling that is not necessarily negative.”

“My instinct is always to seek out challenges as opposed to avoiding them.”

“I read Hemingway, Dostoevsky, Hesse, Camus, and Jack Kerouac.”

“…numbers to leave numbers, or form to leave form, I am describing a process in which technical information is integrated into what feels like natural intelligence.”

“The game had become endlessly fascinating to me, and its implications stretched far beyond winning and losing – I was no longer primarily refining the skill of playing chess, but was discovering myself through chess.”

“A key component of high-level learning is cultivating a resilient awareness that is the older, conscious embodiment of a child’s playful obliviousness.”

“After finishing On The Road, I began reading The Dharma Bums, Kerouac’s fantastic story centering on the Beat Generation’s relationship to Zen Buddhism.”

“…the essence of Tai Chi Chuan as a martial art is not to clash with the opponent but to blend with his energy, yield to it, and overcome with softness.”

“If aggression meets empty space it tends to defeat itself.”

“Investment in loss is giving yourself to the learning process.”

“They were locked up by the need to be correct.”

“For much of this book I have described my vision of the road to mastery – you start with the fundamentals, get a solid foundation fueled by understanding the principles of your discipline, then you expand and refine your repertoire, guided by your individual predispositions, while keeping in touch, however abstractly, with what you feel to be the essential core of the art. What results is a network of deeply internalized, interconnected knowledge that expands from a central, personal locus point. The question of intuition relates to how that network is navigated and used as fuel for creative insight.”

“…chunking and carved neural pathways. Chunking relates to the mind’s ability to assimilate large amounts of information into a cluster that is bound together by certain patterns or principles particular to a given discipline.”

“By ‘carved neural pathways’ I am referring to the process of creating chunks and the navigation system between chunks.”

“…and much of what separates the great from the very good is deep presence, relaxation of the conscious mind, which allows the unconscious to flow unhindered. This is a nuanced and largely misunderstood state of mind that when refined involves a subtle reintegration of the conscious mind into a free-flowing unconscious process. The idea is to shift the primary role from the conscious to the unconscious without blissing out and losing the precision the conscious can provide.”

“Our realities are very different. I am ‘seeing’ much more than he is seeing.”

“…performance training center called LGE in Orlando, Florida. LGE (recently renamed the Human Performance Institute) was founded by Loehr, the esteemed sports nutritionist Jack Groppel, and the no-nonsence physical trainer Pat Etcheberry as an environment in which the physical and mental sides of the pursuit of excellence converged.”

“…if I was doing 3 sets of 15 repetitions of a bench press, I would leave exactly 45 seconds between sets. If I was doing 3 sets of 12 repetitions with heavier weights, I would need 50 seconds between sets, if my sets were 10 reps I would take 55 seconds, and if I was lifting heavy weights, at 3 sets of 8 reps, I would take one minute between reps. This is a good baseline for an average athlete to work with.”

“For beginners, this meditation may seem frustrating because they notice their minds racing all over the place and feel that they are doing badly; but that is not the case. The return to breath is the key to this form of meditation.”

“At chess tournaments, I would meditate for an hour while listening to a tape that soother me, and then I would go to war.”

“I had learned from Jack Groppel at LGE to eat five almonds every forty-five minutes during a long chess game, to stay in a steady state of alrtness and strength. In martial arts tournaments, I now tend to snack on Clig Bars, bananas, and protein shakes whenever necessary.”

“…anger. As we enter into this discussion, please keep in mind the three steps I described as being critical to resilient, self-sufficient performance. First, we learn to flow with distraction, like that blade of grass bending to the wind. Then we learn to use distraction, inspiring ourselves with what initially would have thrown us off our games. Finally we learn to re-create the inspiring settings internally. We learn to make sandals.”

“I had to learn to use my moment organically. Instead of being thrown off by or denying my irritation, I had to somehow channel it into a profound state of concentration.”

“I believe that our world is destroying itself with a cycle of violence begetting violence, and I don’t want to have any part in that cycle. I first got involved with Tai Chi Chuan as a movement away from ego, away from fighting. I was drawn to the experience of harmony and inteconnectedness that felt like a counterpoint to the dog-eat-dog chess world.”

“It is easy to speak of nonviolence when I am in a flower garden. The real internal challenge is to maintain that fundamental perspective when confronted by hostility, aggression, and pain.”

“The first step I had to make was the recognize that the problem was mine, not Frank’s. There will always be creeps in the world, and I had to learn how to deal with them with a cool head.”

“His tactics didn’t touch me emotionally, and when unclouded, I was simply at a much higher level than him.”

“This involved some technical growth, and in order to make those steps I had to recognize the relationship between anger, ego, and fear.”

“The only way to succeed is to acknowledge reality and funnel it, take the nerves and use them. We must be prepared for imperfection.”

“Guys like Miller, Jordan, Hernandez, and Robinson are so far beyond shakable that opponents, instead of playing mental games, cower for fear of inspiring them.”

“Of course there were stages to this process. As a teenager I was thrown off by emotion and tried to block it out. Then, in my early twenties, during my initial experiments with Buddhist and Taoist meditation, I worked on letting my emotions pass like a cloud. This was interesting as it opened up a working relationship with my emotional reality very much like how I described working with the unconscious in the chapter Slowing Down Time. Instead of being dominated by or denying my passions, I slowly learned how to observe them and feel how they infused my moment with creativity, freshness, or darkness.”

“We are built to be sharpest when in danger, but protected lives have distanced us from our natural abilities to channel our energies. Instead of running from our emotions or being swept away by their initial gusts, we should learn to sit with them, become at peace with their unique flavors, and ultimately discover deep pools of inspiration.”

“First, we cultivate The Soft Zone, we sit with our emotions, observe them, work with them, learn how to let them float away if they are rocking our boat, and how to use them when they are fueling our creativity. Then we turn our weaknesses into strengths until there is no denial of our natural eruptions and nerves sharpen our game, fear alerts us, anger funnels into focus. Next we discover what emotional states trigger our greatest performances. This is truly a personal question. Some of us will be most creative when ebullient, others when morose. To each his own. Then Make Sandals, become your own earthquake, Spike Lee, or tailing fastball. Discover what states work best for you and, like Kasparov, build condensed triggers so you can pull from your deepest reservoirs of creative inspiration at will.”

“I have talked about style, personal taste, being true to your natural disposition. This theme is critical at all stages of the learning process. If you think about the high-end learning principles that I have discussed in this book, they all spring out of the deep, creative plunge into an initially small pool of information. In the early chapters, I described the importance of a chess player laying a solid foundation by studying positions of reduced complexity (endgame before opening). Then we apply the internalized principles to increasingly complex scenarios. In Making Smaller Circles we take a single technique or idea and practice it until we feel its essence. Then we gradually condense the movements while maintaining their power, until we are left with an extremely potent and nearly invisible arsenal. In Slowing Down Time, we again focus on a select group of techniques and internalize them until the mind perceives them in tremendous detail. After training in this manner, we can see more frames in an equal amount of time, so things feel slowed down. In The Illusion of the Mystical, we use our cultivation of the last two principles to control the intention of the opponent – and again, we do this by zooming in on very small details to which others are completely oblivious.”

“The beautiful thing about this approach to learning is that once we have felt the profound refinement of a skill, no matter how small it may be, we can then use that feeling as a beacon of quality as we expand our focus onto more and more material. Once you know what good feels like, you can zero in on it, search it out regardless of the pursuit. On a large scale, this is how I translated my understanding of chess to the martial arts. On a smaller, more focused scale, this is how I trained for the 2004 World Championships.”

“When I think about creativity, it is always in relation to a foundation. We have our knowledge. It becomes deeply internalized until we can access it without thinking about it. Then we have a leap that uses what we know to go one of two steps further. We make a discovery. Most people stop here and hope that they will become inspired and reach that state of ‘divine insight’ again. In my mind, this is a missed opportunity. Imagine that you are building a pyramid of knowledge. Every level is constructed of technical information and principles that explain that information and condense it into chunks (as I explained in the chapter Slowing Down Time). Once you have internalized enough information to complete one level of the pyramid, you move on to the next. Say you are ten or twelve levels in. Then you have a creative burst like the ones Dan and I had in the ring. In that moment, it is as if you are seeing something that is suspended in the sky just above the top of your pyramid. There is a connection between that discovery and what you know – or else you wouldn’t have discovered it – and you can find that connection if you try. The next step is to figure out the technical components of your creation. Figure out what makes the ‘magic’ tick.”

“Josh Waitzkin, an eight-time National Chess Champion in his youth, was the subject of the book and movie Searching for Bobby Fischer. At eighteen, he published his first book, Josh Waitzkin’s Attacking Chess. Since the age of twenty, he has developed and been spokesperson for Chessmaster, the largest computer chess program in the world, currently in its tenth edition. Now a martial arts champion, he holds a combined twenty-one National Championship titles in addition to several World Championship titles. When not traveling the country giving seminars and keynote presentations, he lives in New York City.”

How To Be Good At Lots Of Stuff

1. Luck – Start Young
Ideally, your parents would have started interacting with you constructively before you had any idea that you were alive: Mozart on the swollen baby-mama belly, face-in-face moving talking and playing like an obsessive parent, encouraging curiosity and exploring, treating child with respect, multiple environments and quiet time to digest the day.

If you’ve already been born and mama didn’t play mozart while you were chillin in the amniotic jacuzzi, no worries, just start now.

2. Efficiency – Train Smart
If you want to be good at something, learn about it, break it down, understand the pieces, and learn appropriately for your goal. If you want to know how to play piano, don’t learn a song, if you want to know one song on piano, don’t music theory.

3. Discipline – Be Consistent
In almost all cases, it is far more beneficial to train regularly than in binges. I’d suggest that practicing piano for 5 minutes in the morning and night has been far more beneficial than for 2 hours twice a week. Do the math; it also saves time for lots of other stuff.

Training regularly trains your body, your muscles, your synapsis, your mood to grow for a certain purpose. Once you feel yourself growing appropriately, learn to add variety to your training to strengthen and accelerate through the learning curve while not leaving yourself in the dust.

4. Perception – Internalize Abstract Lessons
You will never be good at lots of stuff by memorizing. You might be great at something, but that’s another post for another day. To be good at lots of stuff, it’s important to be able to apply lessons learned today to new situations tomorrow.

5. Conclusion – Bringing It All Together
Now that we’ve reduced practice time by Training Smart, we can further exploit Abstraction by incorporating it into our training. By training generally to run and jump and throw in a variety of ways, instead of how to throw a frisbee, run downhill, and jump out of planes, we prepare by arming ourselves with the potential to be good at many things, instead of just pulling disc, out-running an avalanche, or tempting the parachute gods.

Formal Definition of Random Variable

Let (Ω, ℱ, P) be a probability space, and (E, ℰ) a measurable space. Then an (E, ℰ)-valued random variable is a function X: Ω→E, which is (ℱ, ℰ)-measurable. That is, such function that for every subset B ∈ ℰ, its preimage lies in ℱ: X −1(B) ∈ ℱ, where X −1(B) = {ω: X(ω) ∈ B}.

(Wikipedia: Random Variable: Formal Definition)

Thought I’d share one the few things that brought a smile to my face today. I love and miss math.