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? 🙂

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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.

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Archimedes: The Father of Mathematics

Summary

  • Birth: c. 287 BC in Syracuse, Sicily (colony of Magna Graecia)
  • Death: c. 212 BC in Syracuse, Sicily (during the Second Punic War)
  • Alias(es): Archimedes of Syracuse
  • Ethnicity: Greek
  • Residence(s): Syracuse, Sicily; Alexandria, Egypt (during school)
  • Language(s): Works were written in Doric Greek (Sicilian)
  • Religion(s): Judaic Christian
  • Father: Phidias/Pheidias (astronomer and mathematician)
  • Mother: Unknown
  • Spouse(s): Unknown
  • Children: Unknown
  • Relatives: King Herion II (unconfirmed), Gelon (unconfirmed)
  • Acquaintances: Conon, Dositheus, Eratosthenes, Heracleides
  • Class/Wealth Notes: Upper
  • Institutions/Degrees: The School of Alexandria
  • Profession(s): Mathematician, engineer, astronomer, physicist, inventor
  • Field(s) of Study: Hydrostatics, Mechanics, Geometry, Calculus, Defense
  • Famous Works: The Sand Reckoner, On the Equilibrium of Planes, On Floating Bodies, On the Measurement of a Circle, On Spirals, On the Sphere and the Cylinder, On Conoids and Spheroids, The Quadrature of the Parabola, Ostomachion, The Method of Mechanical Theorems, Book of Lemmas (Liber Assumptorum), Cattle Problem
  • Legacy: “Eureka!”; known as “The Father of Mathematics”; with Newton and Gauss he is commonly referred to as one of the three greatest mathematicians who ever lived; last words were “Do not disturb my circles”;
  • Cause of Death: Killed in Syracuse, Sicily during the Second Punic War despite orders from the Roman general Marcellus to leave him unharmed. The Greek historian Plutarch reported that Roman soldiers killed Archimedes to steal his scientific instruments. Another version states he was stabbed for ignoring a Roman soldier’s orders because he was too entranced in a geometrical diagram he drew in the sand.
  • Notable Historian(s): Isidore of Miletus, Eutocius, Plutarch, Polybius, Thābit ibn Qurra (Arabic translator), Gerard of Cremona (Latin translator)


Archimedes’ Principle & The First Law of Hydrostatics

Story: Archimedes was tasked to determine if the new crown made for King Herion II was made of solid gold. While taking a bath, he observed the level of water in the tub rise as he got in… leading to his “Eureka!” moment regarding density and displacement.

Science: A body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. Therefore Archimedes could immerse the crown in water, measure the amount of water displaced, divide it by the weight of the crown, and arrive at the density of the crown.

Impact: Hydrostatics, or the study of the mechanical properties of liquids at rest, was born. Archimedes’ Principle regarding buyancy and density is used throughout science today. It’s used in the building of ships, other industrial manufacturing, and really any type of engineering. Without it, well, we might be “screwed” (see Archimedes’ other works below).


Other Works

  • Archimedes’ Screw – This consists of a long screw enclosed in a cylinder. With tilted so that its bottom tip is placed in the water, turning the screw pushes water up the screw and out the top. This was used to bilge water out of large ship he designed, the Syracusia.
  • Law of the Lever – Achimedes supplied the first real scientific explanation of how levers work in his work titled On The Equilibrium of Planes (although he certainly did not invent levers).
  • Method of Exhaustion and Pi – Archimedes used the “method of exhaustion” to determine approximate areas and volumes of circles. It involves drawing one polygon outside of a circle, and inscribing a similar polygon on the inside of the circle. Since the area of a polygon (at that time) could be worked out more easily than a circle, Archimedes would determine the areas of the polygons, continuously adding more sides to the polygons, computing the new areas, and estimate the area of the circle which falls between those of the inner and outer polygons. This helped him determine an approximation of pi which he set at somewhere between 3.1429 and 3.1408.
  • Spheres and Cylinders – Archimedes, through the use of several means, proved that a sphere had two-thirds the volume and surface area of a cylinder that circumscribes the sphere.
  • Engineering Feats – Archimedes engineered and built several machines, based on the physical properties and relationships he had proven, to help defend Syracuse from the Roman assault. These included giant pulleys and catapults that would lift ships out of the water and shake them up, destroying them (check out the “claw of Archimedes”). He also built a giant mirror that focused the sunlight onto a ship to burn it.


Adsideological Discussion

Archimedes’ life highlights when a needs translates to accomplishments. This is a characteristic of most inventions, because they need money to flourish and inventors need money to succeed and continue inventing. But Archimedes’ accomplishments were much more than this. It seems to me that he was driven by pure curiosity and intellect, a desire to test his mind against science and nature.

At some level, perhaps he spent too little time outside of his passion of mathematics and discovery. A passion is supposed to be a majority consumer of time and energy. However, no legacy really exists, outside of his scientific accomplishments, that tells us about Archimedes the man and Archimedes the neighbor. Perhaps this has something to do with the time frame in which he lived, but a story told is a story told. Regardless, Archimedes was a life changer and contributed an incredible balance of both an immediate impact and a long term impact on society.

Everything Is Connected

Whether it’s love and hate, birds and weather, past and future, or me and you, there are connections – both hidden and in plain sight – in everything. More than ever, we are finding that the world is a web, and I’m not just talking about the internet. That being said, the internet does help us bring some new connections to the surface through data sharing, communication, and information retrieval.

Math is a valuable support mechanism for these types of connections, especially when credible data exists that is representative of both sides of the river. It often can build the bridge to connect the shores, although it cannot always build traffic between the two.

I’ve posted before on the connections of seemingly unrelated phenomena. How can we determine where connections should (and should not) exist? How can we determine the strength and impact (both direct and potential) of such connections? What are the implications of humans controlling such connections and manipulating the bare characteristics by which some things are connected? These are questions to which we may never have an answer, but it’s important to at least ask the questions and attempt the answers. You never know where a new bridge might appear.

Whether its physical, metaphysical, mathematical, sociological, technological, chemical, theological, biological, philosophical, etc. the connections do exist. To start, we know scientific law covers the physical: Newton’s Law of Universal Gravitation tell us that every object in this universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. For the others, well, let’s just say the bridges are infinite and are always under construction.

on circles: simplicity and perfection

Within the bounds of modern human cognition, the circle is the most basic – but also the most perfect – shape.

I really enjoyed (and highly recommend) the recent New York Times piece titled “The Circular Logic of the Universe” by Natalie Angier. In this article, Angier makes reference to circles of both natural and man-made origin, discussing their physical characteristics and their often debated meanings.

Angier also uses the Russian artist and theorist Wassily Kandinsky‘s piece “Several Circles” (image above) as a foundation for her article. Wassily Kandinsky found interest in the abstract and concrete nature of shapes, delicately balancing geometry as a science and an art.

“The circle is the most modest form, but asserts itself unconditionally. It is simultaneously stable and unstable, loud and soft, a single tension that carries countless tensions within it.”

“The circle, is the synthesis of the greatest oppositions. It combines the concentric and the eccentric in a single form and in equilibrium. Of the three primary forms, it points most clearly to the fourth dimension.”

Some of this stuff is pretty wild if you take a moment and think about it. Circles are everywhere and have been researched for millennia. The earliest known use of the wheel was around 3500 B.C. My current reading “Of Men and Numbers” by Janet Muir talks at length about Archimedes’ (and others’) mathematical discoveries and engineering feats regarding the circle well over two thousand years ago! Fast forward and we are still researching and finding new meaning in circles today – in engineering, cosmology, social science, mathematics and every other knowledge branch.

I guess it’s safe to say that new questions will continuously arise for which we may never find an answer – for all things, circles included. No matter how simple they seem or how perfect they look, the wheels will always go round and round.