the ln of x is no more complex, than cooking or tying a shoe

just find the power to which e should be raised, to equal the x in your queue

now e may seem quirky, this two point seven one thing

but it’s just another constant, with an infinite decimal string

# Tag: math

# World Statistics Day and the Importance of Statistics in Government

The most recent issue of Amstat News features a wonderful summary of the first ever World Statistics Day, which just occurred on October 20, 2010. The article features a series of quotes from the chief statisticians at various U.S. government agencies, all of which serve as a great overview of the critical importance, broad applicability, and growing need for statistics and statistics professionals in the U.S. and around the world. Collectively, we must embrace not only the numbers, data, methods, analyses, and reports, but also the conversations and the debate around such components. In a world heavily fueled by data, I’m very glad that statistics is gaining more international awareness and recognition so that all our lives can be bettered by more informed decisions and debates.

*“Statistics produced by the federal government inform public and private decisionmakers in shaping policies, managing and monitoring programs, identifying problems and opportunities for improvement, tracking progress, and measuring change. The programs of our statistical system furnish key information to guide decisionmakers as they respond to pressing challenges, including those associated with the economy, agriculture, crime, education, energy, the environment, health, science, and transportation. In a very real sense, these statistics provide data users with a lens to focus the myriad activities of our society into a more coherent picture of the status, progress, and trends in our nation. The ability of governments, businesses, and individuals to make appropriate decisions about budgets, employment, investments, taxes, and a host of other important matters depends critically on the ready availability of relevant, accurate, and timely federal statistics. Our economy’s complexity, growth, and rapid structural changes require that public and private leaders have unbiased, relevant information on which to base their decisions.”*

– Katherine Wallman, Chief Statistician, Office of Management and Budget and Past President of the ASA

A few more important (and relevant) statistics resources can be found at:

- FedStats.gov – provides access to the full range of official statistical information produced by the Federal Government (html)
- Overview and Coordination of U.S. Federal Statistics System – 2007 (OMB, 2007, pdf)
- Federal Statistics Programs – FY2009 (OMB, 2009, pdf)
- “Top Statistician Explains What All Those Numbers Mean” – Interview with OMB’s Katherine Wallman (ScienceNews, Nov 2010, html)

# Six Sigma, Switching Spices, and Embracing the Slight Deviations in Life

Although in many aspects of life we must minimize variation to obtain desirable outcomes, it’s when we embrace the slight deviations from normalcy that we obtain leverage, advancement, and enrichment.

**Six Sigma**

Six Sigma, developed by Motorola in 1981, is a “rigorous and disciplined methodology that uses data and statistical analysis to measure and improve a company’s operational performance by identifying and eliminating defects.” In other words, it’s a business management strategy that seeks to minimize variation in operational processes to obtain desirable results for that business/industry.

For manufacturing, production, risk management, supply chain management, accounting, customer service, and many other traditional business functions, minimizing variation is critical for ensuring sustainability, accountability, and efficiency. If the outputs of these functions deviated from what was to be expected, well, it could be expected that the people, the business, and the industry could all be severely impacted at some level.

But in order to spur innovation, create new channels for business, and adapt to markets and mediums that are constantly in flux, these businesses must foster and embrace the slight deviations from what is traditional or expected. There are incredible resources available to allow for these deviations to be leveraged without enormous risk to the bottom line, public image, or financial outlook:

- The internet is an amazingly efficient platform to test new strategies, engage with the public, and collaborate with the universe.
- Statistical methods supply new insight to what may have been and what might be, should this or that occur, with one thing or another considered.
- Social networks can be easily tapped and leveraged for business insight. More is understood about behavioral patterns and social networks than ever before, allowing more direct correlation of business decisions to societal impact.

Business functions, organizations, and entire industries can be bettered by embracing and running with such deviations, even if the short-term prospect could be unknown and questionable. Balancing normalcy with cultured variations is a mixed business strategy that provides leverage within that market, advances industry, and enriches society.

**Switching Spices**

Let’s move from biz to grub. Think of cooking as a math problem. Ingredients are your variables/inputs, methods are your coefficients/operators, and your dish is the output. Given the huge number of ingredients and spices, cooking and plating techniques, and methods of consumption, the range of outputs is somewhere around or above infinity. But given that our options are so vast, it’s amazing how much the output might change if just one of our inputs is changed.

The dish is our dynamical system. Sometimes all it takes is turning up the temperature, or maybe adding more juice, or switching a spice, and the dish becomes entirely new. This is math and food in bed together – the application of chaos theory to culinary experience – making slight deviations from recipes and “comfort-zone” cooking to find new dishes worth trying, sharing, and bragging about.

As much as cooking is an experience, it’s also an experiment. There may be structure – in terms of baking methods and recipe books and kitchen etiquette – but in reality, the door is wide open. Ingredients are for the using, and recipes are for abusing. The best dishes are the unexpected ones, the ones that deviated from expectation, the ones that turned from trial and error to don’t-want-to-share. The mistakes are worth making, for it’s the hundreds of bad pasta dishes that lead to the thousands of great ones. Without embracing the variation in cooking, well, we mind as well hook up to the same gas pump each day.

And lastly, if the world of cooking was its own planet, every inch of it would be covered with a different species, color, scent, appearance, and shape. There is an infinite number of combinations of ingredients, quantities, temperatures, styles, and dishes to consume. Sometimes just switching one spice with another or stirring a little less makes all the difference in making your palate happy and opening a world of new potential dishes. Embracing slight variations in cooking will create new kitchen opportunities, expand your breadth of culinary knowledge and experience, and enrich your palate with a vast array of potential flavors.

**Adaptive Normality**

So what would our world look like if everything was constantly normal? Would we even have a concept of normality? With no variation from what has been done previously, we would essentially cease to learn, experiment, discover, and grow as a society and civilization.

*What makes individuals unique makes many individuals stronger.*

Our characteristics give us dimension. Our characteristics – from eye colors to expressions to birthmarks – give us each an identity that we own while making our society as a whole much stronger, multi-dimensional, and poised to grow.

Our choices give us direction. Our choices – from *picking a college to financial spending habits to lending a hand – fuel and steer us down towards success and happiness, down roads that sometimes seem endless, foggy, and even non-existent.*

Realizing that much good in our lives is based on slight deviations from normality, we must continue to pursue opportunities away from the norm. We must adapt our conceptualization of normality from a straight line to one that constantly moves and includes the variation in life. Our threshold for risk must include these slight deviations so we make them a part of our everyday life. Pushing the envelope in multiple ways brings advancement and enrichment. Divergent thinking, trying new dishes, and taking roads less traveled are all small deviations worth embracing. Although it’s normalcy that might keep us standing, its variation that moves us forward.

# Focus, Balance, and Strength

One is for focus, two for balance, and three for strength. From the most basic sequence of integers we can understand critical characteristics and qualities that, in a sense, provide a backbone by which we can be happy, learn, and grow.

One is one. There is nothing to surround it, there is nothing to be bent. It’s the focal point of many, and the starting spot for all. Above one comes everything else and into one everything comes.

Our society puts a lot of focus on one. We like to see a single result and hear a single voice. We want to find our soul mate and discover the holy grail. We seek to structure our world by its basic individual units, the atoms and nodes. We break down our problems into individually digestible chunks. One is the basic unit of math, the center of gravity, the perfect result. One is the focus and concentration of everything else.

But one stands alone. Where one is one, one is only one. One would be none if no two came from one.

Two is the balance of ones, the pairs of nature, the couplets of science, the squares of math, the rhythm and meter of poetry. Two is evenness and congruence. Two is good and evil, hot and cold, yes and no, high and low, winners and losers, protons and electrons, male and female, life and death. From two we can find harmony and bliss and make connections not previously seen by focusing on one. Two is love. Love is two. Two is the threading of life and the creator of balance within the cosmos. Two is the secret order within disorder, through connections and relationships that make us more than one.

But two still lacks shape. Where two is two, there is only one view of two. Two would be one if no three came from two.

Three is the unit of strength, the shape of our space. It represents our current (most common) perception of spatial dimensions. Three is triangulation, inflection, exponentiation, and curvature. Three is the operation and its result – a combination of the whole picture. Threes provide motion and non-linearity, a dynamic quality of life. Threes make twos unique and unbounded while making stronger our threads. Three is two and one together, forging balance and focus for strength.

Three is the strongest number. Geometrically, the triangle is the only shape that cannot be deformed without changing the length of one of its sides. Spatially, three provides dimension and perception. Three is our basic unit of existence and reality, and well, most of our buildings too.

Three also represents complexity in knowledge. If two is the threads, three is the knots. Three is multiple connections – knowledge with shape. Tie two threads together and you’re building new shapes, discovering new binds, making new questions for answers worth seeking.

And triplets are an optimization of our minds. Remember two things and you could have remembered a third. Try to remember four things and you are likely to leave one out. Triplets are an innate unit of the human mind, something by which we are all naturally bound.

**Focus, balance, and strength.** With three we find strength, and from three we derive balance and focus. Three qualities that make us better individuals, partners, and citizens. Three qualities that, if we learn to utilize and optimize through our life, will surely better our professional, personal, and spiritual lives.

And at the end of the day, numbers are an underlying language of life. We can look to numbers to represent many aspects of life – both physical and philosophical – to help understand how we interact, how we grow, and how to succeed. Looking at a simple sequence of numbers can provide insights that are easier to understand in a world of infinite space and color. Numbers help provide shape to our thoughts and can thread our understanding across cultures and generations. Now did somebody say math is boring? 🙂

# Math Tricks, Negative Space, and Simple Beauty

Once again we start with two of my favorite things: soccer and math. I’ve talked about them both at length, for example in my “geometry in soccer” post from March 2009. Both are related by a similar underlying, structured framework. Both have rules, methods, and strategies for finding success, whether that’s solving a problem or winning a game.

What most non-players don’t understand is that despite the rules that govern both math and soccer, there are tricks to the game as well. These are the visions and insights that exist not within the simple rules and methods of an operation or a play, but rather in the negative space – the non-obvious space surrounding the operations and plays. You may find, more often than not, that recognizing these tricks in all aspects of life can provide the competing advantage necessary for happiness and success.

The soccer tricks will have to wait until after some knee surgery, so for now, I’ll stick with the math. There are thousands of known tricks in math, and probably an infinitesimal amount of unknown tricks waiting for an epiphany of recognition. Here’s an example:

**Squaring Any Number Ending in “5”
**

Although this works for any number that ends in 5, it’s probably most practical for two digit numbers when no calculator is present. Let’s use 65 as an example, where we try to quickly compute 65 squared, or 65^2.

All you have to do is look at the number to the left of the “5” in the ones place. For our example, we have a “6”. Multiply this number by the number that follows it sequentially, which is “7” for our example. We get 6*7=42. To find our final answer of 65^2, all we have to do is take the result of our multiplication and append a “25” to the end of it, recognizing that the last two digits of the square of any number ending in “5” will always be “25”.

So for our example, we have “42” + “25” which gives us 4,225. The square of 65 is 4,225. Pretty neat, huh? Try it with some others…

25^2: 2*3=6, “6” + “25” = 625

95^2: 9*10=90, “90” + “25” = 9,025

475^2: 47*48=2,256, “2256” + “25” = 225,625

For a proof, I’ve looked to Dr. Math at the MathForum.org website. Here goes:

Let’s generalize a two-digit number ending in “5” by the representation X5, where X could be 1, 2, .., 8, or 9. Essentially, X5 is really a shorthand notation for the integer represented by

10*X + 5

Let’s go ahead and square X5:

(X5)^2 = (10*X + 5)^2 = (10*X + 5)*(10*X + 5) = 100*X^2 + 100*X + 25

Now factor our the 100 and an X from the first two terms:

= 100(X^2 + X) + 25 = 100*X*(X+1) + 25

Looking at this closely, you can see that this is exactly the product of X and the next sequential integer (X+1) with “25” appended to the end. Pretty cool, huh?

Notice that this trick works for squaring any integer that ends in “5”, not just two-digit numbers. Dr. Math shows us that for the the larger proof would have to be modified a bit (since all integers that end in “5” cannot be represented by 10*X + 5).

**Seemingly Complex, But Beautifully Simple**

Although the rules and structure of math may at times seem complex and chaotic, in the negative space of math we can find a beautiful simplicity through which things can fall in place. The same can be true for soccer, language, love, astronomy, cooking, and all aspects of life. Sometimes we’ve defined a framework (or have had it defined for us) of rules and methods to follow. But if we take a step back, look between the numbers and think outside the box, maybe we’ll find a simpler route to happiness and success.

# Postulating Possible, Probable, and Preferable Futures

With regards to futurology and future studies, I’ve recently posted on the principles of forecasting as well as the origins of opportunity – two distant yet related topics that exemplify the breadth and depth of the field.

As my own futures research has progressed, I’ve found various sources that have proven to be quite valuable in guiding my curiosity and conjecturing. I think some of these are worth passing along:

- Acceleration Studies Foundation (ASF) – ASF is an educational 501(c)(3) nonprofit engaged in outreach, education, research, and selective advocacy with respect to issues of accelerating change.
- Futurology (Wikipedia)
- Institute for Alternative Futures – The Institute for Alternative Futures (IAF) is a nonprofit research and educational organization founded in 1977. IAF and its for-profit subsidiary, Alternative Futures Associates (AFA), specialize in aiding organizations and individuals to more wisely choose and create their preferred futures. IAF works with clients to create forecasts, scenarios, goals and strategies that are the essential tools for transforming organizations to succeed in times of rapid change.
- Institute For The Future (IFTF)
- Principles of Forecasting (ForPrin)
- Shaping Tomorrow – Online community of futurists and futures research
- Shaping Tomorrow (Ning Network)
- “The Time Lords” (Financial Times, 1/30/2007)
- World Future Society (WFS) – The World Future Society is a nonprofit, nonpartisan scientific and educational association of people interested in how social and technological developments are shaping the future. The Society was founded in 1966 and is chartered as a nonprofit educational and scientific organization in Washington, D.C.
- World Futures Studies Federation (WFSF) – The World Futures Studies Federation (WFSF) is a global NGO that was founded in the 1960s to encourage and promote the development of futures studies as a transdisciplinary academic and professional field in all parts of the world. WFSF operates as a global network of practicing futurists – researchers, teachers, scholars, policy analysts, activists and others from approximately 60 countries.

In a nutshell, futures studies is exactly that – studies of the future. It’s about the determining the total set of possibilities for tomorrow, finding the most probable of outcomes for tomorrow, and leveraging advanced knowledge to even shape the future.

Wikipedia, as spread across the spectrum of credibility as it may be, provides a pretty decent explanation of the mindset needed to be truly forward-looking. What qualities are required of a futurist, within any organization or for any requirement or need, to fully anticipate the unknown using the set of resources at his/her disposal (data, intuition, logic, technology, vision, science, etc.)?

- Trend Assessment – The competency to understand trend directions, weak signals and wildcards, assess their likely impact and effect on one another and respond in a timely and appropriate manner
- Pattern Recognition – The ability to see patterns rather than individual factors
- System Perspective: The capability to envision the entire system rather than the isolated components
- Anticipation: To anticipate short and long term consequences over time, novel situations and geography
- Instinct & Logic: To rely on a combination of instincts and logic rather than purely rational analysis

Yes, there is a plethora of philosophical, political, religious, scientific, and even incomprehensible factors that may give shape to the future (or take that shape away). But that should not deter us from facing it head on. We all need to work smarter not harder, avoid surprises, exploit new opportunities, plug weaknesses, and (where possible) influence the future.

# The Power of Anticipation

*In today’s society, gaining an inch can be like gaining a mile.*

Soccer takes a lot of skill and athleticism. You need *to be able to dribble, pass, shoot, tackle, communicate, see, sprint, etc. But as I’ve stated before (“mind bend it like beckham” – 2/11/2009) it’s just as much a mental game as it is a physical one. You need to think like your opponent and play somewhat of a guessing game, connecting dots before there’s any visible relationship between them. You need to forecast outcomes, intellectually seeing into the future guided by the data that’s available.*

This sort of anticipation is an imperative ability for success in the future – within any endeavor. In business, anticipation means a gaining a leading edge on the competition. For defense, it means preparation and contingency plans for what might be likely to occur. In decision-making its gaining threshold confidence in your decision – using as much relevant information to guide a range of actions, opinion,s and ultimately, outcomes. And not to mention, it helps us grab our umbrella when running out the door.

Predictive analytics, although a seemingly new, hot topic today, has been around forever. Prophets, Mayans, Nostradamus, Pythia, lunar calendars, and the Akashwani – in a historical sense the predictions were informed by a variety of sensory stimuli coupled with intuition and a variety of other external factors. Nowadays, it’s really not that different. Today, we have data and semi-sophisticated mathematical processes that parallel conscious perception and intuition. We can quantify much of what could not have been quantified in the past.

“Predictive analytics encompasses a variety of techniques from statistics, data mining and game theory that analyze current and historical facts to make predictions about future events.

In business, predictive models exploit patterns found in historical and transactional data to identify risks and opportunities. Models capture relationships among many factors to allow assessment of risk or potential associated with a particular set of conditions, guiding decision making for candidate transactions.” (Wikipedia)

It’s imperative that people embrace predictive analytics to inform decision-making. Math doesn’t have to make the decision – that’s mostly for humans – but the math can give a comprehensive picture that outlines components of the decision and also tells us what the decision may lead to (or may have led to in the past) in terms of primary, secondary, and tertiary outcomes. Bruce Bueno de Mesquita is a great example of this, using computer algorithms to predict world events of the future – war, proliferation, conflict, etc. Decisions are not made by computer models, but humans are briefed of probable scenarios in order to make better-informed decisions.

I’ve said this before – math can be simple when it’s made to be simple. It’s a toolbox of problem-solving techniques and thought processes to help guide real-world decisions and understanding. It’s important to not be afraid of the math – start small and grow your mathematical toolbox over time. Take it head on and don’t be overwhelmed. We all have something to learn and we all have something to gain by embracing prediction and anticipation.

So whether it’s sport, meteorology, national security, or adding garlic to the pan, find a way to anticipate. In doing so, my prediction is that you’ll be better off…

**Links**

# All About The Number 100

In celebration of my 100th post coming earlier this week, I figured I would discuss the number 100!!! I know, what a way to celebrate…

**Applications**

The number of yards in a football field.

The minimum number of yards for a par 3 hole in golf.

The number of years in a century.

The number of cents in a dollar (or pence in a pound sterling)

The boiling temperature of water at sea level, in Celsius.

The atomic number of fermium which is made by blasting plutonium with neutrons (named after the great nuclear physicist Enrico Fermi).

The number of senators in the United States Senate.

The number of tiles in a standard Scrabble set.

The basis for percentages (100% represents wholeness, purity, and perfection).

In China, tradition holds that the naming of a newborn panda must wait until the cub is 100 years old.

Pythagoreans considered 100 as divinely divine because it is the square (10^2) of the divine decad (10).

Nostradamus’ work titled “Centuries” contains 10 chapters of 100 verses each.

There are 100 squares in the 10×10 Euler (Latin or Graeco-Roman) Square. A Latin square consists of sets of the numbers 0 to 9 arranged in such a way that no orthogonal (row or column) contains the same number twice. See the image above for an example of a colorful Gaeco-Roman Square for n=10 (the capability for which was discovered by E.T. Parker of Remington Rand in 1959, disproving earlier Eulerian conjectures that a 10×10 square was impossible).

**In Language**

“Cem” – Portuguese

“Cent” – French

“Cento” – Italian

“Cien” – Spanish

“Honderd” – Dutch

“Hundert” – German

“Hundra” – Swedish

“Hundre” – Norwegian

“Hundred” – English

“Hundrede” – Danish

“Hyaku” – Japanese

“Miyya” – Arabic

“Sad” – Farsi

“Sada” – Estonian

“Sata” – Finnish

“Sto” – Croatian, Czech, Polish

“Száz” – Hungarian

“Yibai” – Chinese

“Yüz” – Turkish

Note: “Cent” is the largest number in the French language that is in alphabetical order. And funny enough, when you spell out 2*5*10=100 in French, it’s all in alphabetical order too! (deux*cinq*dix=cent)

**A Mathematical Investigation**

100 = 2^2 * 5^2 (factorization of 100)

100 = (1 + 2 + 3 + 4)^2

100 = 1^3 + 2^3 + 3^3 + 4^3 = 1 + 8 + 27 + 64

100 = The sum of the first nine prime numbers (2+3+5+7+11+13+17+19+23)

100 = The sum of four pairs of prime numbers (47+53, 17+83, 3+97, 41+59)

100 = The sum of the first ten odd numbers (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19)

100 = 2^6 + 6^2 making it a Leyland Number.

100 can be expressed as a sum of some of its divisors making it a semi-perfect number.

100 is divisible by the number of primes below it (25) making it a polygonal number.

100 is divisible by the sum of its digits (in both base 10 and base 4) making it a Harshad Number.

100 is the 854th to 856th digits of pi.

100 is the 3036th to 3038th digits of phi.

In numerology, 100 equals “I LOVE WISDOM TRUTH BEAUTY”

(9) + (3 + 6 + 4 + 5) + (2 + 5 + 1 + 3 + 2 + 7) + (5 + 9 + 1 + 4 + 6 + 4)

**Sources / Links**

- The Number 100 (Wikipedia)
- Number Freak: From 1 to 200 (by Derrick Niederman)
- On The Number 100 (Wisdom Portal)
- Euler (Latin) Squares (Wolfram MathWorld)

# The Intersection Of Expertise

As I begin my job search (25 applications in 2 days so far!) I keep asking myself how to describe what I’m looking for in a job and in what realm do I wish to work? There is no specific job title that describes my experience and education (e.g. “doctor” or “software engineer”) and there is no one department in which I’ve worked or wish to work (e.g. “Operations” or “Logistics” ). Yes, I have an academic background in mathematics & statistics yet it’s difficult to communicate why I have that academic background. I do not necessarily want to become a statistician but rather I fully understand the quantitative nature of things and the power that numbers, math, and quantitative methods have in all aspects of business, government, and life.

So where does this leave me? Well, unemployed and confused, for one. But that’s okay with me. I’m confident that with my capabilities, no matter how hard they may be to communicate in an application or even to a recruiter, I’ll find the position that leverages my abilities and motivation.

That being said, I think I’m at least getting close to describing where I stand, and in real-world terms. It’s at an intersection of sorts – between quantitative methods, scientific and technological realms, and the human element. It’s interdisciplinary – can fit within any group or team or stand alone as an independent researcher or consultant. It’s also dynamic – parallels the speed with which modern business operates and the flexibility required to optimally support the needs and requirements of many types of personnel.

I’ve used a similar image a few times, in posts on knowledge innovation and math in 2010 and beyond. Here I’ve intersected three main topics while including some of my strengths in the middle. Now if I could only match those to a job title…

At what intersection do you operate?

# Boundaries Of The Human Condition

*“That ideas should freely spread from one to another over the globe, for the moral and mutual instruction of man, and improvement of his condition, seems to have been peculiarly and benevolently designed by nature, when she made them, like fire, expansible over all space, without lessening their density at any point, and like the air in which we breathe, move, and have our physical being, incapable of confinement or exclusive appropriation.”* – Thomas Jefferson

There exist many concepts and rules by which we are bound, some of which we may be aware and some of which we may not be aware. Those concepts and rules of which we are aware exist throughout nature and space because we can observe them and learn them, manipulate them and control them, and hear them and speak them. Those concepts and rules of which we are not aware exist because we cannot observe them and learn them, manipulate them and control them, and hear them and speak them. In a sense, we are bounded by that which we can know and cannot know – although those boundaries can and will change throughout the course of history.

It’s interesting to think about our intellectual boundaries, limits, and intersections because they can be sliced and diced a thousand and one different ways. To a chef, his or her capacity may be bound by a colander, letting some things in and others out, clogged and dirty at times and crystal clear at others. To a biologist or chemist, he or she may see it as some semi-permeable membrane that expands and contracts, filters substances based on the needs of the whole system. And to an astronomer, the boundaries may be the vast unknown of our universe: with new discovery always comes more knowns coupled with more unknowns.

Regardless of the profession, it’s valuable to think about. For me, I’ll gladly wear the shoes of a different scientist each waking day but to start, here are a couple different categorizations of our intellectual boundaries, just to jot some thought.

**Spatial Dimensionality**

Think of our intellectual capacity as bounded by one big room. This room can grow as it’s supported by more material, can shrink with the absence of structural connections, and can lose energy with a loss of insulation, cracks in the windows, etc. It can become more complex or simple in a hour’s time with the addition or removal of new features and can take on a new look and persona with the manipulation of a few simple characteristics such as paint and fixtures. You get the point.

*Walls* – The walls are the support and protection, and are the primary means by which we are bound. The walls are our rules of lateral movement, being, and knowing. In a room of infinitesimal walls, we’ll find just as many corners (getting us ever close to the perfect circle) but we’ll still be limited by a surrounding perimeters. In our room, the walls are our physical concepts, our school subjects, our theorems and laws, our rules of society.

*Floors* – The floors are our foundation. Without the floor we would not be able to maintain our position and as a result, move from one position to another. The floors are our foundation for thought – our family, our circumstance, our physicality – our reference point.

*Ceilings* – The ceiling is our limit. The ceiling provides cover and security, shape and reflection, and a foundation for belief and new thought. The ceilings are our hypotheses and conjectures, our gateway to the unknown as much as it they’re the gateway for belief and clarity of vision.

*Corners* – The corners are the intersections of life, the crossroads of knowledge and new thought. Every corner is formed by the other structures mentioned above. The corners are the relationships, the interdisciplinary nature of life, the idea that everything is connected.

**Existential Dimensionality**

Now think a bit differently. Think that our intellectual capacity is bounded by core concepts which, when intersected, form feelings, thoughts, beliefs, and understanding. The core concepts are the things we should study – the basics of existence from which we should gain our foundations. I spoke about studying people earlier, with an overview of Archimedes. For the places, I’ll talk about some of my 2010 visits in the near future. And for time, we’ll it’s the scale by which we can make sense of history, and the perception and reasoning that comes with it. The triangulation of these three things gives an enclosure of feelings, thoughts, and beliefs that form the boundaries of our intellectual capacity.

*People* – We are who we are as much as we are who we’re with (and who used to be with us). To feel, learn, and think, we must understand how other people feel, learn, and think (or felt, learned, and thought). This is core to society, law, science, religion, and everything else.

*Places* – We are who we are in the place that we are. If I were in a different place right now, my actions, feelings, thoughts, and beliefs may be different as a result. Place is a part of circumstance which most certainly contributes to our thoughts and beliefs.

*Time* – We are who we are because of the historical context in which we live. Time forms this context and provides structure to the way we think, how we can act, and as a result, what we might think and believe.

*Feelings, Thoughts, & Beliefs* – Our coordinates at any one time (say, x=people, y=place, z=time) describe who we are. The result of who we are is an output of feelings, thoughts, and beliefs. These form the boundaries, limits, and intersections of our intellectual capacity. Change coordinates, and we’ll find new outputs. And the most important thing to note: as with mathematical coordinate systems, there’s no limit to our coordinate system space, only to a local solid surrounding a group of coordinates. Limits may exist on my axes, by not on the coordinate system as a whole.

**Links**