Nine Basic Shapes Of Life - Common Denominators

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In science, we are always looking for rules and "common denominators" which allow us to understand how things happen (a reverse engineering process), allow us to build or duplicate things, and allow us to predict things -- sometimes to intervene in events and to actually shape the future instead of passively trying to predict it. Fans of The Braintenance Blog are the most curious of all.

 All forms of life are constructed of nine basic designs. The nine designs are listed below and are often combined as composite forms that can include all nine or any combination of several.  From a probability and possibility standpoint, this is fascinating and vast. The number of possible combinations is unfathomably enormous, yet finite.  Here is a listing of these fundamental shapes:

1. the sphere and spheroid
2. the circle and ellipse
3. the cube
4. the cylinder
5. the spiral
6. the undulate
7. the pyramid and triangle
8. the lattice
9. the frond

Identifying the world's shapes can be a fascinating exercise. For example, conifer trees, such as the spruce and the balsam fir, illustrate how objects in nature can combine all of the forms.

• The tree itself is a pyramid (7).
• Its root system and boughs are arranged in dendritic patterns and fronds (9).
• Roots and boughs spread outward from a cylinder of the trunk (4).
• Around the trunk the branches rise in a spiral (5). (Spirals are common in nature as seen in the vine of a morning glory flower, in the scales of flowers and cones, and in the way leaves and needles wind around twigs and branches.)
• A close examination of a pine tree's needles reveals an undulate (6). In general, leaves have the undulated (i.e., toothed, notched, or wave-edged) pattern.
• The tree's resin ducts, like the minerals Halite (salt) and Galena (lead ore), are cubes (3).
• Its needles, like twigs and human fingers, are cylinders (4).
• Leaf and wood fibers are lattices (8).
• The cells in the wood and needles (like those in blood) are spheres or spheroids (1).
• Finally, if the sun shines just right, the shadow of a tree forms a rough circle or ellipse (2).

When you look at any living thing as a whole, it is made of many geometric shapes or patterns – the composite. We don't need to know scientific names, economic value, or botanical details to appreciate a tree. We can appreciate the tree simply because it is beautiful in many ways. As John Keats wrote:

"Beauty is truth; Truth beauty.

That's all on earth ye know.

And all ye need to know."

~~~

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Braintenance: Sand Sifting and Cypherin'

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Braintenance: Sand Sifting and Cypherin’

Here's a Braintenance question from over a week ago, with the answer posted right underneath it. I’ll bet you’d thought that I’d forgotten about it. Well…I never forget; I just get distracted.

Since we've become much more knowledgeable about cylinders and cones, this question has two parts:

If we wanted to divide a truckload of 100 cubic feet of sand (note that we're working on sand, because it's easier to part the sand than to part the sea, generally speaking) equally into two containers, and one of them is a cylinder with a base (radius) of three feet, and the other is a cone with a radius of nine feet, how tall must the cone be? How tall must the cylinder be?

As we recall:

The Formula for the Volume of a Cylinder = Pi x radius^2 x height [or Pi, times the radius squared, times the height]

The Formula for the Volume of a Cone = 1/3 x Pi x radius^2 x height [or one-third of Pi, times the radius squared, times the height]

Pi is the Greek letter representing the constant ratio between the circumference of a circle and its diameter, which is often expressed as either 22/7, or as 3.14.

To get the answers, we just substitute the radius measurements and the amount of square footage of sand (both of which we’ve been given, thank goodness) into the formulas:

For the cylinder, it will be –
Pi x 3 feet^2 x height = 50 cubic feet, or
3.14 x 9 x height = 50, or
28.26 x height = 50, or
Height = 50/ 28.26 = 1.769 feet in height. (a short, squat cylinder…kind of stumpy, in fact)

For the cone, it will be –
1/3 x Pi x 9 feet^2 x height = 50 cubic feet, or
1/3 x 3.14 x 81 x height = 50, or
84.78 x height = 50, or
Height = 50/ 84.78 = 0.589 feet in height (a short, squat cone…even stumpier than the cylinder)

Observation = The height of the cone is one-third the height of the cylinder when the radius of the cone is three times that of the cylinder. Is this reciprocal relationship always true? Yes it is. In every case where the relationship between the cone's radius and the cylinder's radius is 1 to 3 (or 1:3), the ratio of the heights (assuming that they each contain the same volume of sand, or something more interesting) will be the inverse, or 3 to 1 (or 3:1).

My, but this gets me tired. I'm leaving. See you soon.
Faithfully,

Douglas Castle
  

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Braintenance: Sorting Sand Into Two Containers, Continued! (02.16.2010)

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Dear Friends:

Here is the question (if you have a negative outlook, you might call it a "problem" instead of a question) which I had posed to you last week: 




The mission is to find the heights of each two containers, one which is cylindrical and one of which is conical such that each one can hold the same exact amount (volume) of sand. Each of the containers must have a floor area (which is a circular base, naturally) of three square feet.

Here are the formulas required to either a) answer the question, or b) solve the problem. These formulas are part of the study of solid geometry:

Quick Note:  In the formulas that follow, instead of using superscript ("powers") to show exponents, I have used the ^ sign. For example, 5^2 is "five to the second power" or "five squared".

The Formula for the Area of a Circle = Pi x radius^2 [or Pi times the radius squared]

The Formula for the Volume of a Cylinder = Pi x radius^2 x height [or Pi, times the radius squared, times the height]

The Formula for the Volume of a Cone = 1/3 x Pi x radius^2 x height [or one-third of Pi, times the radius squared, times the height]

Pi is the Greek letter representing the constant ratio between the circumference of a circle and its diameter, which is often expressed as either 22/7, or as 3.14.

ANSWER (SOLUTION)

The first thing we know is that both containers have the same sized base (a circle, or more properly, a disc) and therefore have the same radius.

Just by observation, it is apparent that a cone having the same radius as a cylinder, would only hold (or contain) one third the amount of anything (such as sand) that it was filled with as would the cyclinder. Put another way, the cylinder can hold three times as much sand as the cone (assuming that they both have the same radius, which also means that they would have the same base size, or footprint). Seen from yet another perspective, the cone would have to be three times the height of the cylinder in order to hold (or contain) the same amount of a substance; in this case, to hold the same amount of sand.

The answer (or the solution) is that the height of the cone must be three times the height of the cylinder. For example, if the cylinder were three feet in height, the cone would have to be nine feet in height; if the cylinder were five feet in height, the cone would have to be fifteen feet in height.  There is an infinite number of possible correct numerical answers, providing that the cone is always three times as tall as the cylinder.
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Here's a question for next time. Since we've become much more knowledgeable about cylinders and cones, this questions has two parts:

If we wanted to divide a truckload of 100 cubic feet of sand (note that we're working on sand, because it's easier to part the sand than to part the sea, generally speaking) equally into two containers, and one of them is a cylinder with a base (radius) of three feet, and the other is a cone with a radius of nine feet, how tall must the cone be? How tall must the cylinder be?

Take good care of your brain. Remember, your brain is like a muscle -- it grows stronger with exercise. Give it a workout. Take it out for a spin. Your mind can expand to meet most any challenge. Ironically, most of us are far more intelligent than we think we are.

Faithfully,

Douglas Castle



About Douglas Castle 
Douglas Castle - LinkedIn Profile
The National Networker Companies
Braintenance - Stay razor sharp.
The Internationalist Page - A world without barriers.
The Global Futurist - Revealing trends.
Taking Command! - Mastering your fate.
Follow Castle on Twitter 
Follow TNNW on Twitter
Follow Braintenance on Twitter
*Subscribe (free!) for The National Networker Newsletter and the BLUE TUESDAY REPORT, and join The TNNWC GICBC at Join Us!