Light, Life and Information

Seeing Clearly

Millions of living things in the animal and plant kingdoms have a working relationship with light.  Any with sentience "know" what that relationship is.  Or, at least, they make decisions based on perceptions of the benefits of that light.

Light is a form of energy as is evident in the growth of plants.  Without light, green plants die.  Even with plenty of nourishment and hydration, the healthiest of plants need light.  This energy is used to combine water, carbon dioxide and various other nutrients to build stems, limbs, leaves and fruit.  This energy is then released when that material is consumed by other organisms or by combustion.

But the most remarkable thing about light is that it allows sentient beings to gather information from the environment remotely.  Direct contact is not necessary when we animals can see things around us.  Some of us need help as our eyes, wonderful light receptors that they are, become weaker.

I know that light moves so fast that, comparatively speaking, things close by and far away are equally accessible.  However, there are ways to slow light down just a little.  When this happens, light is scattered, absorbed or reflected.  In this way, over time, scientists have figured out that light travels through a vacuum at about 186,282 miles per second.  That is fast enough to travel around the earth at the equator about 7.5 times!

Since light moves so fast, information from all over the universe is accessible on a clear night.  Certain stars, along with the moon, allow navigation along the surface of the earth because the information is predictable.  Based on the apparent movement of the sun on the horizon at sunrise and sunset, we are able to discern the passage of time.

So, what do I know?

• Light is basically energy.
• Light travels very fast.
• Light enables access to information.

Falling is Normal

I move from basic math concepts which I "know" by instinct to other things that are harder to prove, but still just as much of my knowledge base.

Let's move to "gravity."  I won't get into the math behind it, for that is in the territory into which geniuses dare to tread!  But I will speak to the concept.

I know that things FALL.  How much more basic can you get?  If there is any space at all between my hand and the floor, when I let go of an object, it falls to the floor. I've heard, and I've seen video evidence to prove it, that all objects fall at the same speed in a vacuum.  That is convenient for making calculations if one knows the math.

Related to gravity is "weight."  This is the evidence of what in science is called "mass" and "density."  The fact that a feather takes longer to fall to the ground that a marble is because the air around us has both mass and density.  Because of this, we have weather as the air is heated to change its density.

Personally, I experience gravity's effect in whatever I am doing.  As I sit here typing this blog, I can feel my body pressing against the chair.  If I am not careful, my legs will grow numb based on pressure distributed upon my hips.  The keyboard rests securely upon the flat surface of the desk, making typing possible.  All thanks to gravity.

If I were to go outside to play a game of catch with my grandson, gravity would be there to challenge me.  I would take advantage of the fact that the ball is of such a mass and density as to be able to temporarily "defy" gravity due to the energy imparted to it by my releasing it in a forward motion.  That motion, though, would diminish with distance as gravity worked by the laws of motion. The ball would either end up on the ground or in the hands of my grandson.

In all likelihood, my grandson's return of the ball would take a curved line.  This line would put the ball further from the ground for a moment.  The arc it forms obeys the laws of gravity and I have to adjust to intercept the ball.

So, What do I know?

Things fall until they hit a surface.

Things have mass and density.

Things can "fly" when energy is exerted against them that is greater than the gravity that makes them fall.

Circular Reasoning

In some ways the circle is more basic to universal knowledge than the straight line.  More specifically, the curved line is most often incorporated in the art of toddlers given a crayon.  The youngster will rarely draw straight lines, but he will draw rough circles (ovals) as he doodles.

When he gets curious, he "looks around" and "goes in circles" to take as much of his world in as he can.  The space near him is "around" him and he is "surrounded" with things in every direction. To take it all in he goes in "circles."  The word "round" descends from the Latin word "rota," from which we get "rotate."  The word "circle" comes from the Latin "circulus," the diminutive of "circus" which it borrowed from the Greek "kirkos" which means "ring."

Even in ancient times, the Romans and Greeks built stadiums that were semi-circular or fully round (oval tracks).  They were built in such a way as to give a  view to as many people as possible.  And so, they built "circles."  The methodology naturally would have begun with the architect stretching a measuring line to the optimum distance and rotating the desired arch -- all the way to a circle.  On paper, the "circus" was a lot smaller, and thus a "circulus."

This line became known as the radius.  The resulting line forming the ring turned out to be about six and a quarter times radius (a bit over 6.28).  The radius, when doubled, became the diameter of the circle. The ratio between the widest part of the circle and the distance around it is an irrational number known as "pi" estimated as "3.14" or "22/7".  The area inside a circle is pi times the square of the radius.

The most ancient system of measuring the outside of a circle used what we call quadrants: the four directions measured by fixed points.  The common points were the sunrise and the sunset at the equinox.  The ancient Hebrews used East as forward, making that which was on the left "north" and tha which on the right "south."  Behind them was the unpassable sea, the "west" and sundown.  This way the circle has been divided into 16 common directions we know today.  However, this is not precise enough over long distances, so "degrees" of the arch use "base 60" with 60 times 60, or 360 degrees divided by "minutes" and "seconds."  For convenience, time also follows this scheme.

By the way, 360 is the result of multiplying 3,4,5 and 6 together. Seen as prime factors, this is 2*2*2*3*3*5.  This makes the factors of 360 a long list: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 54, 60, 90, 120 and 180.  All in all, the ancient math guys had it together.

So, What do I know?

The radius is a straight line used from a center point to inscribe a circle.

The diameter of a circle is twice the length of the radius.

The distance around the circle is about 3.1416 times the diameter and about 6.2832 times that of the radius. This is the ratio known as "pi".

The area inside the circle is pi (about three and a seventh) times the square of the radius.

A Square Deal

Geometry 4:

A Square Deal

When working with the right triangle, it is easy to build a "rectangle" by simply building a mirror image of the right triangle.  One might expect the term "right rectangle" to be used here, but that would be redundant.  This is because the prefix "rect" is a corruption of the original Germanic form of the Latin "rectus" which means "right."  The Old German was "reht" but the "h" was the hard "kh" sound that migrated up from the Greek "Chi" (looks like our "X").  The Old English form was riht, corrected (pun intended) by adding back in the hard sound.  This time, the hard "g" was used, rendering "ri-ght"  It did not take long before the hardness of the "chi" sound was abandoned.  From "rect" to "right" and back again!

Anyway, we have seen how a right angle can be constructed using the 3:4:5 ratio with the sides.  Using the largest side (the diagonal "5") a ratio of 5:4:3 becomes the mirror image forming a 3 by 4 rectangle.  The area of the rectangle is simple to determine in "square" units.  The area of a "right quadrangle" (four-sided with parallel sides) will be twice that of a right triangle.  Another way of seeing this is that the formula for triangle will be one half that of a rectangle.

The formulae for this are simple:

Area of a rectangle = base (width) x height (length).
Area of a triangle = 1/2 x base x height

The "square" is a special kind of rectangle.  Not only are all the angles inside a square equal, but so are the sides.  The ratio of the sides of the "half square" loses its whole number diagonal when "side a" and "side b" are forced to be the same size.  The diagonal becomes a multiple of the square root of 2! That works out to a little over 1.414.  I'd go out farther, but fourteen-fourteen works for me. But, when considering the area of the square, no irrational numbers need be involved.  I really prefer rational numbers.

When we move to other triangles and polygons, where the angles are not all right (no pun intended), then the "height" becomes a problem.  The height of a triangle is always figured in a right angle, which must be used to divide the shape in order to follow the rules of the square.  That is to say, everything must be reduced to right triangles and/or rectangles to be squared.  It's like playing with blocks!

I'll work out the bugs of those building blocks in another blog.  I know just enough geometry to be "dangerous."

So, what do I know?

Rectangle: Area = lw
Rt. Triangle:  Area = (1/2)lw = lw/2, where l and w form the right angle.

Square: area = l² (length squared) [length being equal to width)

All area within polygons are measured by the pythagorean theorem:  a² + b² = c²

The Triumphant Triangle

Geometry: Part 3

The Triumphant Triangle

As mentioned last time, in order to "square up" a patio, you have to make a triangle.  Of course, this shape is named by the number of angles it has.  After a straight line between two points, the next thing is getting to a third point.  The shortest path between the three points forms a three-sided enclosure with three angles: the "tri-angle."

The prefix "tri" bypasses the introduction of the "th" (theta) that replaced the original "t" (tau) in English.  The vowel sound changes depending on the context and language, but the original Indo-European "trei" is preserved in the Greek treis and its cognates in Western languages. From all I could find out, its meaning has always been "three" (2 + 1).

Though there seems to be a dualism in much of nature (good and bad, light and darkness, wet and dry, and so forth), there is also a triad of moderation that adds depth to reality.  These three dimensions define reality: past, present and future; liquid, solid and gas; width, length and height.  It is the latter of these that is of interest to us right now.

Without a third point, no shapes can be formed. This is true in plane geometry as well as solid geometry.  It is in two dimensions, at a time, that basic shapes are made on a flat surface.  On that surface, a triangle is the most stable of all shapes.  This is because once the three sides are connected, the angles are fixed in place.  To change an angle, one of the sides has to be compromised.  The changing of the length of a side changes the angles as the sides are reconnected.

Going back to the right triangle, the one with the set ratio of 3:4:5, let us say that side 'a' (3) is shortened to 2 units (let's use meters).  To maintain the right triangle, the side opposite the right angle now must be shortened as well.  The new length will be the square root of 20 (a little less than 4.5 meters). To save the 5 meter pole, all the angles change as well.  Rotating outward to about 108 degrees, the 5 foot side once again secures to an stable, though offset triangle. At a ratio of 2:4:5, the ratio is definitely not "right."

And now, let me show how the sum of the three angles will always be twice that of a right angle.  The reason I don't give this in degrees is because the presently defined circle of 360 degrees is based on assuming the "base 60" system of antiquity.  It makes dividing a circle into even numbers quite easy, but is not "known" in all cultures.  For the record, 360 is 12 times 30, or 3 x 4 x 5 x 6.

So, a right angle was formed halfway along an arch which will make a straight line.  If you take the original length of cord - that is 3 meters - you can make a semi-circle defined by a straight line.  This forms the base for two identical triangles, each with the ratio of 3:4:5.  Let's take the cord that forms the original triangle and move the second peg (point b) that was three meters out.  Moving it further from point c, the new position of the peg increases the angle of angle BCA while reducing the other angles.  All along, the length of the cord remains twelve (3 + 4 + 5) meters long.  The area inside the cord diminishes toward 0 until the two halves of the cord reach a length of 6 meters.

The cord is now twice the length of the perpendicular line that formed the right angle and the angle of that line is also twice that of the right angle.  At all times, the angles inside the changing triangles added up to twice the right angle.  If the peg had move toward point c, at some point the sides would have become equal, with each angle also being equal, or one third the angle of a straight line. In this case, 3 sides of 4 meters each.

So, What do I know?

A triangle is defined as a shape on a plane which has three lines intersecting at three angles.

The angles within any triangle will add to that of a straight line.  This total will be twice the angle formed by a perpendicular line bisecting that line.

In a triangle with three equal angles, the sides will be of equal length.  The angles will add up to that of a straight line.

In mathematical terms:

Right Angle BAC = 1/2 x (Angle ABC + Angle BCA)

Triangle ABC = Angle BAC + Angle ABC + Angle BCA

Given Right Angle BAC = 90 degrees, line segment BC will have 180 degrees.

Getting It Right

Geometry: Part 2

The Right Angle

In order to make sure your patio is squared, whether it is going to be a rectangle or a square, you will
need the cord three times the length of the distance to the distance to the first peg.

You really don't even have to measure the distance to the first peg to assure a square corner, but having a tape measure will make this easier.  Let's say your patio is to extend 12 feet (or, if you'd rather go metric, 4 meters).  This means you will need 36 feet (or twelve meters) of cord to easily square up your patio.  The cord will be a little bit longer to leave room for securing it to the pegs.

As pictured in figure #1, the tripling of the length is done by walking back and forth between the pegs with the spool of cord.

Once you have the full length (#2), walk that back to the first peg (#3) and then repeat (#4).  In the end, the four strands will be 9 feet (3 meters) long.  If you began against a wall, just lay the 9 foot piece against the wall and put the third peg down.

Fasten the 9-foot length to the third peg and the end of full cord to the first peg.  Taking the full cord, walk to the second peg, moving it to where it provides a taut line between all the pegs.

This works because in every triangle with a right angle the side opposite that angle has a length that is the square root of the sum of the squares of the other two sides.  It so happens that this ratio is found first in whole numbers with 3, 4, and 5.  And so, any multiples of these numbers produces similar triangles.  If you are going to be doing a lot of building, you could make your own "square" tool using a yardstick or even a twelve inch ruler.  Or, of course, just buy one when you get a chance!

So, what do I know?

A right triangle can be constructed based on the formula

a² + b² = c²

That is, when side c is opposite the right angle, the squares of sides a and b add up to the square of side c.

This manifests itself the ratio of 3:4:5.  That is to say, 9 + 16 = 25.  In whole numbers this only works with multiples of these three numbers.  The ratio using 1:2 would render an irrational number: the square root of 5.  Likewise 2:3 would need the square root of 13!  It only works with these three adjacent whole numbers!

Drawing the Line

Geometry: Part 1
Drawing the Line

We saw the "line" earlier, when we put numbers on it. I was working on an assumption that I knew what a line was.  But this is a blog about what I know, and I have laid down numbers about as far as I can go, so I will be moving on to geometry, the science of construction.

On a flat surface, the straight line is the shortest distance between to places.  On the surface of a sphere, that line is necessarily curved in relation to the radius of that sphere. I will begin with flat surfaces and straight lines in what is called "plane geometry."  That is not "plain," but "plane," as in flat.  It is what we usually think of when building things anyway.

I should begin with the zeroth dimension, which is the point.  A point has no dimensions, but it has coordinates.  The best way to envision a point, though, is as a very small circle, just big enough to see.  That way, we will know where we are and can see where we are going.

To make this practical, as a Greek guy named Euclid did, we will envision laying out a patio.  Making the sides straight will not be hard, and the next blog I will deal with making it rectangular -- or at least triangular, which is exactly half of a rectangle.  But first, the line.

It is quite easy to draw a straight line.  With nothing but a few pegs, a mallet and a measured cord, a perfectly straight line will be a cinch.  After placing a peg in the first corner, measure out a length of cord to a multiple of 4 that is equal to, or beyond the length of the patio.  Do not cut the cord, for you will need twice that much more to assure a square corner.

* * * three pegs
and a cord
---------------------------------


But first, the straight line must be drawn.  With one end secured to the first peg, pull the cord taunt. With a second peg connected to the end of the cord, etch an arch of maybe step in each direction, tentatively marking the middle of that arch.

If the cord is tight, you have your first straight line.  It is the shortest distances between the chosen points.  The slight curved line that you have made in the sand is the beginning of a big circle with a radius the length of that line, but that will come in handy later.

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So, what do I know?

A straight line on a plane is the shortest distance between two points.  The easiest way to make a straight line is with a string, cord or rope.  Even with modern tools, this is sometimes faster.

A point in space has no dimensions, and a line is in one dimension on flat surface in two opposite directions.

Next: A right triangle.