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.

People Studying People

Society is people. Whether it’s business growth, intellectual advancement, government, mass media, artistic culture, knowledge transfer, sports successes, health care, economic development, or charity, it all starts with people. Therefore, in order to learn about how society is shaped and how it can change, it’s imperative that we learn as much as we can about the people who have come before us.

One purpose of my blog is to organize information about influential people of the past and present to try and pass this information on to others. Adsideology very much follows this notion – that life is about people, and we should study people to become people. I do recognize that the more diverse the people, the more wholesome the information gained. However, I’ll probably start with some mathematicians as I’ve recently bought a few books on the great ones in history.

As a start, let’s think of some numbers. Solely focusing on Earth, how many people have ever lived? Most estimates fall around 100 billion total. The interesting note about this number is that with a current population of over 6.7 billion, this means that almost 7% of people ever born in the history of Earth are living today. In other words, only 93% of people ever born have ever died! Pretty wild to think about, right?

On Knowledge Innovation

I want to quickly mention a correlating note regarding knowledge innovation for the future – how new thought can best be stimulated given the current state of society. It is clear that one pillar of innovation will always be people – the human component. Even in a world growing in reliance on information services, the human component will always remain. I’ve posted previously on the need for the human component in future mathematics initiatives as well as the need for expanded human intervention for optimized search technologies. The fact is, the human component will always be there. Common sense, yes, but commonly understood, maybe not.

Links

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.

a simple estimation of height

Geometry is useful for more than just passing the sixth grade.

In October, I posted on estimation as an essential analytical tool to have today (and more importantly, tomorrow). It’s useful for scheduling, planning, purchasing, and other decision-making circumstances. Well here’s a quick and easy geometric technique for estimating the height of very large things. All you need is an intermediate height of reference (perhaps a friend) and your eyes.

For this example, I will use a friend as my intermediate point of reference and a large building as the object for which I wish to estimate the height.

Line up your friend between you and the building. Your friend should be positioned so that when your eyes (A) are as close as possible to the ground, the top of your friend’s head (C) lines up with the top of the building (E). You’re essentially creating the hypotenuse of a large triangle!

Now, let’s label and identify the other parts of our picture.

Here are our labels.

Given this picture, geometry tells us that certain relationships exist.

Therefore, three easy estimations must be made in order to get the estimated height of the building (y):

       w = the distance between you and your friend
       x = the distance between you and the building
       z =  the height of your friend 

NOTE: Be sure to use the same units in your estimations (feet or yards, perhaps) or else your calculation will not work. 

Once you have those three values, just leave the rest to geometry. You have basically created one right triangle inside another right triangle, assuming the building and your friend are both standing up straight. Therefore they have equal angles and therefore equal ratios of their legs, allowing us to make this simple calculation. The result:

Math is fun, right? 🙂

perfects, primes, and planets

a simple poem with a significant end
get the result, and perhaps you’re a friend
so riddle me this, riddle me that,
use pen and paper and your best thinking cap…

let’s start you off, here’s one for the money
in chinese it’s death, unlucky but funny

the seventh fibonacci should be easy to see
but if that’s too hard then take two to the three

now keep the dwarf in the orbital loop,
give me a count of our planetary group

a champion of numbers, a winner of sort,
when one has won and they’re top of their sport

then something to fathom, the star is divine
or give me a width of a vitruvian kind

now how bout a gimme, a favorite of mine
this one’s the one only even prime

and last but not least, i hope you’re awake
the first perfect is this keplerian snowflake

now put them together, from bottom to top
a day in my life – big thanks, mom and pop!

the mind as a map

The human mind should work much like modern mapping and camera technology – zoom, pan, adjust, layer, interact – and export too.

At any moment, the majority of minds fall into one of two categories: big and strategic, or focused and tactical. But as changing times require changing minds, the third category has emerged: the dynamic and balanced. This category can be seen as a mix of the first two, instantaneously being able to function based on the attributes of the surrounding medium.

These minds are very much like new cameras, mapping applications, GPS tools, and related emerging technologies. They build a informative picture for a user, based off organized databases and knowledge bases, and allow a level of functional interaction to continuously feed new information to that user. These functionalities, when applied to the human mind, are all essential for continued growth in a rapidly changing (and unpredictable) society.

Zoom

  • Act as a lens. Be able to zoom in and out from a single focal point. For any given topic, the mind must be able to pay attention to the smallest of details while still being able to see the big picture. Understand the color and shape of the individual puzzle pieces while at the same time seeing where that piece fits into the full picture on the puzzle box.
  • Re-focus instantaneously at every level of zoom. Purposely making pictures blurry can provide useful in some instances, but the act of focusing should be natural and automatic. 
  • Like looking at a Magic Eye or a lovely Seurat, be able to find the right level of zoom where the picture is most clear.
  • “Zoom Analytics” as I’ll call it, should be embraced as a common analytical method. It’s always been a mathematical problem solving technique, but not universally taught.

Pan

  • Need to be able to swiftly move from topic to topic, and connect those that are related.
  • Moving back to a previously-visited topic should bring quicker loading of that memory.

Adjust

  • The mind must continuously grow in dimension and adjust for core characteristics. Recognize patterns and contrasts, shapes and sizes, color and form and adjust the view and output accordingly.
  • Toggle perspective and angle to see the infinite sides of any one picture. Perspective is everything.

Layer

  • If the brain consisted of data and memory silos, the main interface should be able to integrate any combination of data and memory into a single comprehensive picture.
  • It should be able to see localized data as well as aggregate data for larger constructs. Filter data and memory based off a set of parameters, re-organize it, and feed it into the common operating picture.

Interact

  • The picture is not static. The brain must by dynamic in nature, allowing a constant influx of new information and updating of old information. 
  • Re-organization of data and memory should be consistent with the changing society in which we live. When a scientific/technological revolution occurs, the way in which our information is processed and stored must be compatible with the changes in society.

Export

  • Not every tool can do every task. That’s why exporting is good. Create a new data set from which you, or someone else, can work. Export a map or a picture that can be analyzed by another set of eyes. For the human, you must be able to transfer stored information to others, and most importantly, communicate it effectively. English is English, math is math, kml is kml.
  • Language is good because it is a standard by which we can effectively communicate. Choosing words wisely is something that should be practiced on top of a common linguistic standard. It’s one thing to speak the same language, but another to foster understanding.

And so, truly finding a balance between big and small perspectives is important. It’s important for making wise decisions, being a team player, being an effective manager, giving valuable advice, and finding optimal direction in life. So as much as you make sure you can get deep in the weeds, make sure you can easily get out.

“It’s not what you’re looking at that matters, it’s what you see.” – Henry David Thoreau

math in 2010 and beyond

If we want to fuel future growth and innovation in mathematics, three worlds must meet in the middle.

In 2009, we see three distinctly developed worlds:
  • The Communities: Math + People = Associations, Publications, Journals, Groups, Departments (ASA, IMS, WFU Math, etc.)
  • The Connectors: People + Technology = Social Media & Social Networks (Facebook, LinkedIn, Twitter, iPhone Apps, etc.)
  • The Foundations: Math + Technology = Software/Web Applications (Wolfram|Alpha, SAS, R, Matlab, Mathematica, Statistica, etc.)

In 2010, we need these three worlds to mold into one, unified experience. With whom does the responsibility lie and when does it start? You and now.

shapes and squiggles

Armed with a pencil and paper, you can simplify about 99% of the world’s problems.

Despite a couple decades of extra-substantial technological growth, there are two things that can never be replaced: the pencil and paper. For the toughest analytical challenges, only so much can be done computationally to simply and digest such problems. For these challenges, the solutions should start with a pencil and paper.

The first step in breaking down a problem is the conversion of the problem from the brain’s three dimensional space to a the two dimensional space of paper. In mathematics, there are several examples of such similar breakdowns: matrix decompositions, polynomial factorizations, projections, transforms, etc. The breakdown is necessary to see things in a new light, a simplified light, and a light that otherwise may not have been turned on.

Step 1. Grab a pad of paper. Do not put boundaries on where you can write and draw.
Step 2. Grab a pencil. Sharpen it and keep the pencil sharpener close.

So now that we have pencil and paper in hand, what do we draw? Well here’s my point. There is a geometric toolbox that provides a valuable framework for the problem solving environment. These are the shapes and squiggles.

1. Matrices

Two-by-two matrices are especially valuable for initial sorting of qualitative data. Assign a binary variable to each axis, name the cells, and define the relationships. Categorizing concepts and attacking each cell independently can help find hidden relationships and provide insight for subsequent analyses. See my previous post on matrix power for more on matrices.

2. Graphs

For more quantitative and scaled concepts, draw a set of axes to start. Visualize relationships between variables by drawing lines or curves and then attack each extremum and graphical sector. Plot knowns and/or hypotheticals on the graph and decipher the meaning of specific coordinates. Jessica Hagy’s blog ‘Indexed’ is a good example of translating mind to graph.

3. Lists and Mind Maps
The proper organization of information is often the most valuable visual tool in solving complex problems. Of course there are technologies to assist in the visualization and organization of information (mind maps, spreadsheets, etc.) but it’s important to use pencil and paper as the primary stepping stone to using some software/web app. Check out a mind map on different mind mapping software and a post on five great uses of mind maps.

4. Circles

Circles have shape and have a shape that is unique. They overlap well, fill space comfortably, and are easy for the human mind to spatially interpret. Eulerian circles (or Venn diagrams) are the simple example of circles put to use on paper for analytical means. There are several other adaptations of circles for comparative reasoning, such as with GL Hoffman’s “gruzzles”.

5. Doodling

The mind works in mysterious ways. Drawing without bounds can release otherwise inexpressible thought. There’s the somewhat structured doodling such as with UI mock-ups, schemas, and decision trees, and very unstructured doodling that might look like an impossible maze of dots and lines. The importance lies in the fact that your brain knows most about the problem, and the pencil is driven by the brain. Any new representation put forth on paper, by your brain, is a new representation of that problem not previously seen. In other words, “doodling allows the unconscious to render in symbolic expression”.

The shapes and squiggles live on. And the shapes and squiggles will always live on because they are the simplest yet most powerful functional tools our mind can use to express our conscious, subconscious, and unconscious thoughts.

matrix power

How much of your life can you fit into rows in columns? Well, enough of it for you to cherish the matrix as a valuable organizational and analytical tool.

Spreadsheets, tables, and matrices are used in every aspect of life. We track finances, monitor tasks, plan our future, and analyze potential relationships with rows and columns. And we are surrounded by this information as individuals, as part of small social groups, and as part of large organizations such as classes, companies, or governments.

More simply, matrices and tables give a new structure to elements of our life that are not always so two-dimensional. From the new structure, we can glean new insights and inspire new visualization of those same elements to make best-informed decisions. To me, a matrix is a valuable analytical tool that helps organize information for insight and action.

(Note that I am using the term “matrix” to represent that much more than numerical arrays of the math world. I am including categorical mappings, tables, lists, and spreadsheets too.)

The University of Cambridge Institute for Manufacturing nicely defines the matrix as an essential decision support tool:
  

“A two by two matrix is a useful tool for initial sorting of qualitative data. The axes should be chosen so that, e.g., the data with the most desirable characteristics will fall into the upper left quadrant and the least desirable in the lower right quadrant. While groups may be unable or unwilling to assign absolute values to qualitative data, they usually find it relatively easy to come to a consensus as to which quadrant something belongs in.

Generally, the two by two matrix is a useful tool for categorising things that can be reduced to two simple variables, particularly when quantitative information is unavailable and qualitative judgments must be made.

It enables a rapid clustering (or separating) of information into four categories, which can be defined to suit the purpose of the exercise. It is particularly useful with groups as a way of visibly plotting out a common understanding or agreement of a subject.”

Authors Alex Lowy and Phil Hood describe the matrix as “the most flexible and portable weapon in the knowledge worker’s intellectual arsenal”.

What’s best about the matrix is that flexibility. Depending on need, you can get as much power out of a 2×2 matrix as you can from a 5×5 matrix. Increased dimension does not translate to increased power. The matrix is flexible and dynamic to the needs of your analysis. You control the path to discovery.

And although matrices do a nice job of pairing categorical relationships, you can also translate these pairs to numerous other visualizations to better contextualize the information at hand. Turn your row and column headers into scaled concepts, map them to some x- and y- axes, and try and fit your qualitative information to a line that describes the relationship between x and y. Is the relationship directly proportional, inversely proportional, linear, parabolic, or along some other path? What do each of these types of relationships mean for your categorical variables?

It’s important to note that there can be fuzzy lines too. Not all cells need to have values and not all relationships need any sort of defined continuity. Empty cells and undefined relationships provide insights that are just as valuable as the populated and defined ones. Lack of data is data in itself, and that’s a great thing.

In the end, the matrix is just one part of the analytical toolbox and can provide a wide range of insight for your personal and professional life. Box up your data, organize it, visualize it, and use new structure to optimize your life.

Examples

Business/Leadership: Gartner, an IT research and advisory company, has created the “Magic Quadrant” to analyze types of entities in the business world. By plotting the ability (or inability) to execute against the completeness (or incompleteness) of vision, businesses can be categorized with those sharing similar characteristics, as Leaders, Challengers, Visionaries and Niche Players. This is a useful example of turning abstract qualities into groups for targeted strategy and decision making.

Product Development/Management: For analyzing how to grow a business from the product side, one matrix shows how plotting types of markets vs types of products can help guide that growth strategy.

Math/Statistics: Type I and Type II error tables are used to describe possible errors made in a statistical decision process. This is a great example of mapping relationships between categories, naming the cells, and using the matrix to understand what each cell represents.