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| Joined: | Mon Nov 21st, 2005 |
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I cannot attest to the intended meaning, all I can do is work on the meaning I work out as those words work in my thinking.
The example here and now has again to do with the math problem concerning surface area to volume relativity, or ratio, and so my thinking turns back to the specific problem I found in words as follows:
Again:
In other words:
Surface area is measured in 2 dimensions and it is therefore a theoretical type of measure, a fictional type of measure, it is a measure that has no depth.
Here:
http://www.ucadia.com/uca/u05/050400.htm
Concepts of dimension (meaning?) is offered in that link; and I roughly connected midstream in that stream of meaning, so as to roughly connect into the specific words specifying the meanings offered concerning dimension.
Back to my thinking:
Surface area as a shape, such as a ball, a globe, an orb, is measurable in 2 dimensions, having no thickness, but it is not simply fictional in fact, it is simply fictional in theory; meaning it has no depth, it is a theoretical amount of space taken up, taken from space, and therefore the obvious thinking, the obvious meaning, is to look deeper into how much space is taken up by this shape, not merely the surface area, but there is space taken up in 3 dimensions, which can then be called volume.
Surface Area = no depth = only shape, or skin, or no thickness
Volume = the total amount of space including depth taken up or displaced, or removed, or occupied, or no longer space, this area, this volume, this place, at this time, is no longer space, this area, this total area, is now occupied by this volume of stuff as this stuff displaces whatever was (or was not) occupying that space before the volume was taken up with new stuff.
I use my imagination at that point and imagine the Moon around the Earth on a toggle switch whereby the one who switches the toggle switch on places the Moon where it is and the same one who switches the toggle switch off takes the Moon out of that space: on, off, on, off, and it may be a 2 year old kid who is at the switch while everyone else on Earth has to deal with the reality of having the Moon exist in that space, then not exist in that space, then exist in that space.
A 3 year old has access to the switch that places and takes away the Earth, and then a 4 year old has access to the switch that places and takes away the Sun.
All three kids have a party and they take away and replace the Moon, Earth, and the Sun in apparent random order: just having fun.
What happens to all the people on the surface of the Earth when their Earth is switched off for that moment when the 3 year old hits the off switch?
The volume and the surface area accurately measure something real, a globe, a planet, but the surface can be taken away at a minimum level, a very small thin, very thin, microscopic thin measure of thickness. The thickness of the surface area can be so extremely thin as to take a kid at a switch taking away one layer for billions of years and at that time, billions of years later, none of the people on Earth can measure the thickness of Earth diminishing at all.
The kid taking away the volume in one instant takes away a lot.
The kid taking away a very small layer of the surface for billions of years and each second of time the kid takes away many layers at a frantic pace as the kid pushes the remove button, removing a thin layer each time, and the layer is so thin that no one on Earth notices any change, at all, in the surface area of Earth. The volume does not change at all, as far as anyone can tell, ever.
So my thinking was along these lines where 1 unit of volume exists as 1 unit of volume, and the shape is a globe, or ball, and then another amount of volume is suddenly placed as a new layer of skin around that 1 unit of volume.
Now there are two shapes to measure.
A.
1 Unit of volume in the shape of a ball.
B.
The same amount of volume as A; but this new shape is a skin that is rapped around A, so this new shape is an inner skin boundary and and outer skin boundary.
A and B are the same amount of volume.
A and B are Globes taking up the same amount of space.
A does not capture any space inside of A.
B captures space inside of B.
A fits exactly into B without changing shape.
B cannot fit into A unless B changes shape.
The math problem is as follows:
What is the thickness (skin) of A compared to the thickness (skin) of B?
The skin or thickness of B is measurable as length from inside surface area to outside surface area: a straight line; in theory a straight line if not precisely straight when measuring with the truest, straightest, possible accurately measuring device.
Thickness of A is Radius not diameter; because it can be noted that the center of A is a very, very, very, very, very, very small globe, or ball, and A is a skin around that ball, so the thickness of A is Radius, not Diameter.
The math problem is then:
Is the radius (thickness) of A larger or smaller than the thickness of B?
I think, not confirmed, that the answer is that A is larger than B in thickness; which is intuitive, or rational, or reasonable, since the same volume as A is outside of A and so that volume occupies more space, so that UNIT of volume is spread thin around the same UNIT of volume.
Seeing this with math rather than with words is worth doing; but I have chores to do and I am still thinking about it, with words, with imagination/imagining shapes.
The first UNIT of volume becomes smaller in thickness when the volume is doubled.
The first time, the first case, the first expanse, of the first UNIT of volume, when that first UNIT doubles is a resulting thinner skin...
So, in time, with that above in mind, the skin thins when doubling volume, so when will the doubling of volume end up with (how many double expanses in time) a very small increase in skin compared to volume?
This is along the same lines as the rate in which the rate of acceleration increases?
During errands the intuition takes over and the increase in size of volume (from 1 Unit upwards, always doubling volume) never changes ratio - it is always the same ratio of Volume to Surface area.
Working with 1 Unit (doubling volume to 2 Units one outside the other as skin) is the same as working with 10 units, 100 units, or 1 billion to the billionth exponent Units, the ratio is always the same, intuitively, because all the observer is doing is adding ZEROS to the process of what happens when volume is doubled.
So that is as far as my intuition goes...
What about the actual math?
Needed is a formula for the volume of a sphere.
http://www.mathsisfun.com/geometry/sphere.html
There is the original problem I had with the words offered by Frank at Ucadia.
Here is Math is Fun:
"Of all the shapes, a sphere has the smallest surface area for a volume. Or put another way it can contain the greatest volume for a fixed surface area."
Here is Ucadia:
"The most efficient shape in terms of number of points combining to create maximum volume is an octahedron (six points) combining to create eight equally proportioned triangles, expanding to a middle point and reducing to a single point. The surface area to volume of a perfect Octahedron is always 1:2. That is, an octahedron creates twice as much volume as it takes surface space to create it. Octahedrons are therefore the simplest and most efficient shapes in terms of minimum number of points for maximum volume creation."
Back to the need for a formula to find volume in a sphere there is this:
Surface Area = 4 × π × r2
Volume = (4/3) × π × r3
I am so ignorant with math, that puts my mind in a tailspin. What is the reasoning for placing (4/3) in the Volume formula? 4 divided by 3 is 1.3333333333333333333
I can check one more source for the Volume equation and then I can start using it.
http://www.wikihow.com/Calculate-the-Volume-of-a-Sphere
V = ⁴⁄₃πr³
I want to start with a unit of 1 Volume so as then to have another (double) of that 1 unit of volume. To get that result the equation is reversed?
http://www.calculatorsoup.com/calculators/geometry-solids/sphere.php
Modern Technology = less knowledge stored in my bio hard drive (brain)?
r = 0.620350491 m
V = 1 m3
A = 4.83597586 m2
C = 3.89777709 m
There it is in 9 decimal places for 1 meter cubed for volume of a sphere.
I am now looking for that same 1 meter cubed volume amount to be doubled and that volume of space will then be a layer of skin around the 1 cubed volume.
r = 0.781592642 m
V = 2 m3
A = 7.67663317 m2
C = 4.9108914 m
Now I want to subtract the 1 m3 radius from the 2 m3 radius
r = 0.781592642 m - r = 0.620350491 m = 0.161242151 m
Now I want to double that volume again, which is 4 meters square; again there is one sphere and it is copied, two identical spheres in volume, but the shape of the next sphere (the double) is placed around the first one.
r = 0.984745022 m
V = 4 m3
A = 12.1858956 m2
C = 6.18733545 m
Again the subtracting of radius from radius
r = 0.984745022 m - r = 0.781592642 m = 0.20315238 m
Now, with math, I can begin to see this phenomenon better in these numbers.
From 1 volume to 2 the increase in radius is compared to the 2 volume to 4:
r = 0.781592642 m - r = 0.620350491 m = 0.161242151 m
r = 0.984745022 m - r = 0.781592642 m = 0.20315238 m
I was not seeing that increase in radius when I tried to conceive the meaning of this phenomenon of increase in volume compared to "taking up space," and now this NEWS that is viewed with math is interestingly confusing.
I think I had thought "intuitively" that the increase in radius would have been constant, not increasing, and then my work with math earlier on changing volume to surface area RATIO, short circuited my thinking into an error of assuming that radius measures from volume doubling would decrease.
So radius measures resulting from volume doubling increase from 0.16 (first doubling from 1 to 2) to 0.2 (second doubling from 2 to 4) instead of staying the same, or decreasing, which should have been intuitive in my thinking, but my thinking is based upon ignorance, mostly, not knowledge.
I am going to double the measure 5 times; and place all 5 in order:
1.
r = 0.620350491 m
V = 1 m3
2.
r = 0.781592642 m
V = 2 m3
3.
r = 0.984745022 m
V = 4 m3
4.
r = 1.2407 m
V = 8 m3
5.
r = 1.56319 m
V = 16 m3
So as to see more clearly the rate of increase in radius:
1. 0.161242151 m
2. 0.20315238 m
3. 0.255954978 m
4. 0.32249
Subtracting those again:
2 from 1 = 0.041910229
3 from 2 = 0.052802598
4 from 3 = 0.066535022
So, thinking in visual increases in sphere volume, while thinking in English language, while looking at numbers. The 1 unit of volume doubles. The radius of 1 unit of volume originally is .62 and then that length from inside to outside of that shape (volume now as skin around the .62 radius) is now .78.
OK, that is where my thinking is again sent off into confusion.
The original 1 unit of volume is measurable as a skin around a smaller volume.
Here is where acceleration, the concept of acceleration, has my brain befuddled.
This is the same problem?
Expressed as:
Acceleration Problem:
Before something starts moving and after something starts moving is a point at which no movement becomes movement.
Occupying Space Problem:
Before something becomes something and after something has become something there is a point at which nothing becomes something and this is being shown with a shape of the thing that becomes something from nothing in the form of a sphere; whereby the center of the sphere is viewable as a unit of measure; or a sphere within a sphere.
In the acceleration problem the movement from point A to point B is realized as an interval of time; such as the Earth is at point A and then at point B as an orbit around the Sun, and that interval of time is 1 year. 1 year is a very long time compared to a very small amount of time. Acceleration MUST be the smallest amount of time possible.
See?
Acceleration is the interval of time required to go from no movement into the first movement from no movement, so that is, in FACT, the smallest measure of time.
That is again this problem of accurately measuring the first displacement of what isn't (nothing) becoming the first accurate measure of what is, such as the first measure of something such as a sphere, a shape, a globe, a thing, an object, where it was, and then it is, again, is the smallest possible accurate measure of it.
Before it can move, whatever it is, it has to be something, and then it moves the smallest increment of movement; much smaller than a second of time, it begins to move, but before it can move, it becomes something that then can move.
So the sphere idea, the shape of it, returning back to my confusion (my thinking was sent into a wobble) with this radius idea.
From the center of what, is measured the radius of a sphere, starting with a unit of measure of 1 unit of volume, which is 1 meter cubed, the radius of that 1 cubic meter of volume is .062 meters of length.
From the center of that 1 unit of volume to the surface of that sphere the length of that center to surface measure is .062 meters.
1 cubic meter of volume is .062 meters of skin surrounding a core of no matter of any kind, because that sphere took away that space.
Again my thinking goes back to the kid at the light switch where nothing was, and then something is, and that something is a sphere of 1 cubic meter in volume and the radius of that volume is .062 meters.
Then the kid doubles the volume but instead of 2 spheres side by side there is now a sphere with .062 meters from center to surface, and outside of that core, there is another thing, a skin, and that skin is measured from inside of it (a volume) to outside of it (a volume) .078 MINUS .062 which is this:
So as to see more clearly the rate of increase in radius:
1. 0.161242151 m THE FIRST DOUBLING OF VOLUME
2. 0.20315238 m
3. 0.255954978 m
4. 0.32249
Original 1 cubic meter volume was: r = 0.620350491 m
Doubling that volume resulted in another thickness which is:
1. 0.161242151 m THE FIRST DOUBLING OF VOLUME
That is because the original 1 cubic meter volume was: r = 0.620350491 m
The total radius of the doubled volume was: r = 0.781592642 m
So the increase in radius from the original 1 cubic meter volume to the 2 cubic meter volume was: r = 0.781592642 m - r = 0.620350491 m = 0.161242151 m
An increase in radius of 0.161242151 m, when 1 sphere becomes twice as much volume.
So as to see more clearly the rate of increase in radius:
1. 0.161242151 m THE FIRST DOUBLING OF VOLUME
2. 0.20315238 m The second doubling of volume
3. 0.255954978 m The third doubling of volume
4. 0.32249 m The fourth doubling of volume
The radius increases as volume doubles.
One unit of volume doubles and the original sphere becomes a skin with an increase of .161 length of its original thickness which was 0.62, becoming .78, which was in increase of .161 length from center.
Was .62 length from center, doubled in volume, becomes .78 length from center, and that is a measurable increase in length from center of .16 more skin around the original sphere.
That can be shown visually with a computer animation.
Still working on this idea of replacing space (what is not) with something such as a sphere the initial thing is what it is and at the center of it is still it, so it is not skin around anything, it is it, and then doubling it's volume, creating a skin around the sphere, is not a smaller amount of skin (original radius reduced in length) the first doubling.
So...the next doubling is an increase (not a reduction) in the length of the skin or depth, or additional length of the total radius from the center.
So...the reverse is then the obvious thing to look at concerning the reduction of radius as the sphere volume is cut in half.
From 1 unit of volume to half a unit of volume the radius of skin reduces. Half the volume is taken away, which is less skin taken away each time half the volume is taken away. As the volume reaches closer to nothing the skin reaches nothing sooner.
I think my mind was confused on that point as the work with volume to surface area ratios appeared to suggest a process that exists universally when no such universal process exists, in other words I made a false assumption during thinking.
I can return to this error precisely as:
Process 1:
__________________________________
Octahedron change in ratio of surface area to volume as size increases:
Unit measure of 1 = Edge
SA for Octahedron A: 3.4641016V for Octahedron A: .4714045333333333
SA / V Ratio = 7.34843900 to 1
Unit measure of 3 = Edge
SA for Octahedron B: 31.1769144
V for Octahedron B: 12.7279223991
SA / V Ratio = 2.449489 to 1
Unit measure of 5 = Edge
SA for Octahedron D: 86.60 cm2
V for Octahedron D: 58.92 cm3
SA / V Ratio = 1.46969 to 1
Question:What happens to the surface area to volume ratio as the Octahedron gets larger?
Answer:SA / V decreases as the Octahedron gets bigger
_________________________________________________
That process is a decreasing of the Surface Area to Volume Ratio as the shape grows larger in size.
I did not do that work for the sphere so that can be done now:
http://www.calculatorsoup.com/calculators/geometry-solids/sphere.php
r = 0.62035 m
V = 1 m3
A = 4.83597 m2
C = 3.89777 m
That is 4.8 Surface area to 1 Volume or 4.8 to 1.
Starting with 1 cubic meter of volume in a sphere the ratio is then:
SA/V Ratio = 4.8 to 1
Doubling the Volume:
r = 0.781593 m
V = 2 m3
A = 7.67664 m2
C = 4.91089 m
That is 7.67664 to 2 or divided (solved) for 1 that is 3.83832 Surface Area to 1 Volume or 3.8 to 1.
So going larger in size the ratio goes from:
4.8 to 1
to
3.8 to 1
That may be the source of my confusion (not hard to do for me and math) as the obvious is a reducing ratio down to nothing in surface area as volume increases.
Following the procedure through with doubling size from 1 to 2 to 4 to 8 to 16:
1
V = 1 m3
A = 4.83597 m2
4.8 to 1
2
V = 2 m3
A = 7.67664 m2
3.8 to 1
4
V = 4 m3
A = 12.1859 m2
3.0 to 1
8
V = 8 m3
A = 19.3439 m2
2.4 to 1
16
V = 16 m3
A = 30.7067 m2
1.9 to 1
100
V = 100 m3
A = 104.188 m2
1.04 to 1
1000
V = 1000 m3
A = 483.597 m2
0.48 to 1
1,000,000,000
V = 1000000000 m3
A = 4835970 m2
0.0048 to 1
So here is the confusion again as the last sphere can be called 1.
The last sphere can be 1 unit of measure.
The smaller unit of measure (1 small sphere) is small and the ratio between Surface Area and Volume is:
4.8 to 1
The larger unit of measure (1 large sphere) is large and the ratio between Surface Area to Volume is:
0.0048 to 1
The sphere grows bigger and the inside volume increases faster than the outside surface area, as a ratio of Volume to Surface Area while the reverse is the inside volume deceasing relative to a faster increasing skin as a ratio of Volume to Surface Area.
Starting with 1 going to .0001, then .0000000001
1
V = 1 m3
A = 4.83597 m2
4.8 to 1
.0001
V = 0.0001 m3
A = 0.0104188 m2
104 to 1
.0000001
V = 1.0E-7 m3
A = 0.000104188 m2
1041 to 1
When surface area is 1,000,000,000,000,000,000,000,000,000,000 to 1 the sphere is big skin compared to little volume relatively speaking?