The Prof
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What Relativity says is, altho we are 3 dimensional beings trapped in 3D perception, space in reality is multi, multi D.
Perhaps infinite D? I believe that's what my math professors at UCB said, literally. That is, IF I understood them correctly.
I don't now if space is truly comprised of an infinite number of dimensions. But I know, for sure, according to the theory that is theory no more, and hasn't been since the 20's---space is certainly multi, multi D.
Many more than 4.
"Space is bent." That's one of the great popularizations, ala the great Carl Sagan, used by those fools, like me, here, who would try to explain that which they can only partly perceive.
A diminsion, in the first place, is A POSSIBLE DIRECTION YOU CAN GO. If you were stuck in one dimension, 1D, ie, a straight line, the only POSSIBLE DIRECTION YOU CAN GO is one---FORWARD. As well as its negation, backward.
Now, if you're graced with an entire 2nd D, then there's TWO possible directions you can go: forward/backward, and left/right.
You're 2D. You're stuck on the floor. On a plane, a 2-dimensional plane.
We, however, are TALL enough to be 3 whole D's. Forward/backward, L/R, and the third direction we're all happy to be able to go---Up/Down.
ANY DIRECTION WE CAN PHYSICALLY POINT TO IS A COMPOSITE OF THOSE 3 COMPONENTS: fwd/back, L/R, up/down.
3 steps forward, 1 step L, 5 feet up.
Or. 10 inches back, 1 inch R, 1 inch down.
Einstein, however, teaches we actually live in more D's than we are able to point.
Ie: there are OTHER POSSIBLE DIRECTIONS WE CAN GO.
We just can't point to em. We just can't PERCEIVE THEM.
And these OTHER directions, these SHORTCUTS in space, have all to do with, for us, HOW FAST YOU GO.
Space is bent.
***
The popularizers of Relativity always begin with the Parable of the Twins.
One rockets thru space NEAR THE SPEED OF LIGHT.
His double remains on Earth.
The space/time traveler returns 1 week later. For him, it's just a week.
But his brother has been dead 10,000 years.
The MEANING of Relativity: WHEN YOU TRAVEL (GO) NEAR THE SPEED OF LIGHT (literally approaching 186G miles per sec):
Time slows down so much it almost stops.
Mass (for us, weight) increases like the inside of a Black Hole.
And distance becomes shorter than Fabe Washingon's attention span.
There are shortcuts thru space. And you find them by going really fast.
The PRACTICAL EFFECT of Einstein: time travel is possible. Into the future, if your vantage point is your point of departure.
Into the universal past, were you to consider all this from that wherever destination you'd be if you went 186G per sec for, oh, even 5 minutes, I'd think.
***
The UCB Math Curriculum, which, c'mon now---when I took those classes all those years ago, man, those teachers up there trying to get thru to ME, some kid---it's all wizardry, necromancy, way beyond my pubescent understandings.
But decades of settling in, and a few key clues, bring perspective. And now I see that what they were pushing were the mathematical foundations underneath Einstein.
And they were 2: Non Euclidean Geometry. And Projective Analysis.
The first takes on Euclid, says he's wrong. Euclid is the Alexandrian Librarian a generation after the great Macedonian conqueror, the geometrist, whose Elements is amazingly, stunningly similar to your old HS geometry text.
Euclid's geometry---point, line, plane, the axioms, postulates, the concept of proof---his 4th postulate violates Einstein.
Euclid's 4th postulate, the Parallel Postulate, says: thru a point OFF a line there is one and only one parallel to the given line.
It's not true. Not in 4D.
Parallel lines meet.
At infinity.
Which you can APPROACH traveling that FAR down one of those lines in a BLINK if you GO almost the speed of light.
Please consider this: This part is difficult, so please slow down a bit.
Consider the Earth, consider the globe.
Please think about the Equator, the ring of latitide encircling the broad girth of the planet like the belt around your waist.
Finally, consider if you will any 2 lines of longitude. Make them neighbors for the sake of ease.
The are BOTH, absolutely, PERPENDICULAR to the equator, both perfect right angles with the longest of the latitudes.
Indeed, they're both perpendicular to ALL lines of latitude.
They are PARALLEL, therefore, any 2 lines of longitude, since they're BOTH PERPENDICULAR TO THE SAME equator.
But they MEET (!!!). At the N pole.
2 parallel lines meet!
Because they are part of a TWO dimensional grid (latitude and longitude) being WRAPPED or BENT around a THREE dimensional object, a GLOBE.
Now, if you can somehow wrap the ENTIRE UNIVERSE around some FOUR D object, whatever the hell that is, which has everything to do with how fast you go, then:
THAT'S IT!
THAT'S IT!
You got it. Relativity!
Which is far as I can take you.
***
Except.
Projective analysis. Sagan opened THIS door for me, certainly not Cal (NUMBER ONE in the nation in undergrad math when I graduated in the muddled middle of my class, 1983).
Please consider: an object and a light source.
Say, a pencil, and any light. But please ANCHOR your lamp in one place, for convenience, direct above your head.
Hold out your pencil and examine its shadow on the floor.
The shape of the shadow changes as you rotate or revolve your little yellow piece of wood.
If you position your pencil parallel to the ground, its shadow is a LINE SEGMENT.
As you rotate your pencil slowly to vertical, your segment CHANGES, shrinks in length.
And if you maneuver it so it's pointing straight down, its reflection is now a little circle, exact size of your eraser.
The 2D representation of the 3D object CHANGES, depending on...
Now, Sagan says we're 3D living in 4D. He compares thus---you see how the SHADOW of a 3D object is always 2D.
Hence, THE SHADOW OF A 4D OBJECT WOULD BE 3D.
WE, us, are shadows of 4D stuff!
Sagan says this: pretend you are a little, flat 2D object. You have length and width but no height. You are a little cutout square of paper lying on the floor.
Now, being a little 2D square you are perfectly free to wander around your little 2D plane, forward/back and L/R to your little papery contentment.
But you can't go up nor down.
You are STUCK on the floor.
You can't even PERCEIVE up and down.
Now, I, Carl Sagan, being 3D, I can look down on you on my floor. I can see you but you, if I pick my feet up off the floor, ie, sit on top my desk, you can't see me.
If I put my feet on the floor, I'D APPEAR SUDDENLY OUT OF TOTAL NOWHERE.
And all you would see is the cross section of my feet, only the length and width of the cross section, you would have no perception of anything like height.
Now, if I walked down the stairs, you would see CHANGING CROSS SECTIONS, 2D slices of my moving 3D presence, changing and transmogrifying in bizarre, cartoony, sci fi fashion---appearing from nowhere of a sudden, and changing insanely, cross sections of my feet, legs, belly, head, then gone.
Poof.
What the hell was that?
***
The math profs will get mad if you say the 4th D is time.
But for our POPULAR understandings of all this, please, the 4th D is time.
The profs would exactly say: time is just one of an infinite number of dimensions beyond length, width and depth.
But, please, prof, it's hard enough as it is.
***
Einstein undoes Newton, the Physics Prophet of his millenium.
They both published each their 3 world changing discoveries at age 26.
Einsteins in the late 10's or early 20's.
He was confirmed in the 20's, I'm pretty sure, when the gravity of the sun was indeed found to BEND light rays from stars just behind our sun, from our vantage.
So close to the sun, those stars, they can only be seen during a total solar eclipse.
When it was found that the stars whose light rays, to reach us, must pass close by the sun, when it was CONFIRMED that those rays did indeed bend, well, Einstein was pretty much proved.
He's been confirmed in all ways since, sub atomic particles near the speed of light, the energy in the atom...
The Prof
Perhaps infinite D? I believe that's what my math professors at UCB said, literally. That is, IF I understood them correctly.
I don't now if space is truly comprised of an infinite number of dimensions. But I know, for sure, according to the theory that is theory no more, and hasn't been since the 20's---space is certainly multi, multi D.
Many more than 4.
"Space is bent." That's one of the great popularizations, ala the great Carl Sagan, used by those fools, like me, here, who would try to explain that which they can only partly perceive.
A diminsion, in the first place, is A POSSIBLE DIRECTION YOU CAN GO. If you were stuck in one dimension, 1D, ie, a straight line, the only POSSIBLE DIRECTION YOU CAN GO is one---FORWARD. As well as its negation, backward.
Now, if you're graced with an entire 2nd D, then there's TWO possible directions you can go: forward/backward, and left/right.
You're 2D. You're stuck on the floor. On a plane, a 2-dimensional plane.
We, however, are TALL enough to be 3 whole D's. Forward/backward, L/R, and the third direction we're all happy to be able to go---Up/Down.
ANY DIRECTION WE CAN PHYSICALLY POINT TO IS A COMPOSITE OF THOSE 3 COMPONENTS: fwd/back, L/R, up/down.
3 steps forward, 1 step L, 5 feet up.
Or. 10 inches back, 1 inch R, 1 inch down.
Einstein, however, teaches we actually live in more D's than we are able to point.
Ie: there are OTHER POSSIBLE DIRECTIONS WE CAN GO.
We just can't point to em. We just can't PERCEIVE THEM.
And these OTHER directions, these SHORTCUTS in space, have all to do with, for us, HOW FAST YOU GO.
Space is bent.
***
The popularizers of Relativity always begin with the Parable of the Twins.
One rockets thru space NEAR THE SPEED OF LIGHT.
His double remains on Earth.
The space/time traveler returns 1 week later. For him, it's just a week.
But his brother has been dead 10,000 years.
The MEANING of Relativity: WHEN YOU TRAVEL (GO) NEAR THE SPEED OF LIGHT (literally approaching 186G miles per sec):
Time slows down so much it almost stops.
Mass (for us, weight) increases like the inside of a Black Hole.
And distance becomes shorter than Fabe Washingon's attention span.
There are shortcuts thru space. And you find them by going really fast.
The PRACTICAL EFFECT of Einstein: time travel is possible. Into the future, if your vantage point is your point of departure.
Into the universal past, were you to consider all this from that wherever destination you'd be if you went 186G per sec for, oh, even 5 minutes, I'd think.
***
The UCB Math Curriculum, which, c'mon now---when I took those classes all those years ago, man, those teachers up there trying to get thru to ME, some kid---it's all wizardry, necromancy, way beyond my pubescent understandings.
But decades of settling in, and a few key clues, bring perspective. And now I see that what they were pushing were the mathematical foundations underneath Einstein.
And they were 2: Non Euclidean Geometry. And Projective Analysis.
The first takes on Euclid, says he's wrong. Euclid is the Alexandrian Librarian a generation after the great Macedonian conqueror, the geometrist, whose Elements is amazingly, stunningly similar to your old HS geometry text.
Euclid's geometry---point, line, plane, the axioms, postulates, the concept of proof---his 4th postulate violates Einstein.
Euclid's 4th postulate, the Parallel Postulate, says: thru a point OFF a line there is one and only one parallel to the given line.
It's not true. Not in 4D.
Parallel lines meet.
At infinity.
Which you can APPROACH traveling that FAR down one of those lines in a BLINK if you GO almost the speed of light.
Please consider this: This part is difficult, so please slow down a bit.
Consider the Earth, consider the globe.
Please think about the Equator, the ring of latitide encircling the broad girth of the planet like the belt around your waist.
Finally, consider if you will any 2 lines of longitude. Make them neighbors for the sake of ease.
The are BOTH, absolutely, PERPENDICULAR to the equator, both perfect right angles with the longest of the latitudes.
Indeed, they're both perpendicular to ALL lines of latitude.
They are PARALLEL, therefore, any 2 lines of longitude, since they're BOTH PERPENDICULAR TO THE SAME equator.
But they MEET (!!!). At the N pole.
2 parallel lines meet!
Because they are part of a TWO dimensional grid (latitude and longitude) being WRAPPED or BENT around a THREE dimensional object, a GLOBE.
Now, if you can somehow wrap the ENTIRE UNIVERSE around some FOUR D object, whatever the hell that is, which has everything to do with how fast you go, then:
THAT'S IT!
THAT'S IT!
You got it. Relativity!
Which is far as I can take you.
***
Except.
Projective analysis. Sagan opened THIS door for me, certainly not Cal (NUMBER ONE in the nation in undergrad math when I graduated in the muddled middle of my class, 1983).
Please consider: an object and a light source.
Say, a pencil, and any light. But please ANCHOR your lamp in one place, for convenience, direct above your head.
Hold out your pencil and examine its shadow on the floor.
The shape of the shadow changes as you rotate or revolve your little yellow piece of wood.
If you position your pencil parallel to the ground, its shadow is a LINE SEGMENT.
As you rotate your pencil slowly to vertical, your segment CHANGES, shrinks in length.
And if you maneuver it so it's pointing straight down, its reflection is now a little circle, exact size of your eraser.
The 2D representation of the 3D object CHANGES, depending on...
Now, Sagan says we're 3D living in 4D. He compares thus---you see how the SHADOW of a 3D object is always 2D.
Hence, THE SHADOW OF A 4D OBJECT WOULD BE 3D.
WE, us, are shadows of 4D stuff!
Sagan says this: pretend you are a little, flat 2D object. You have length and width but no height. You are a little cutout square of paper lying on the floor.
Now, being a little 2D square you are perfectly free to wander around your little 2D plane, forward/back and L/R to your little papery contentment.
But you can't go up nor down.
You are STUCK on the floor.
You can't even PERCEIVE up and down.
Now, I, Carl Sagan, being 3D, I can look down on you on my floor. I can see you but you, if I pick my feet up off the floor, ie, sit on top my desk, you can't see me.
If I put my feet on the floor, I'D APPEAR SUDDENLY OUT OF TOTAL NOWHERE.
And all you would see is the cross section of my feet, only the length and width of the cross section, you would have no perception of anything like height.
Now, if I walked down the stairs, you would see CHANGING CROSS SECTIONS, 2D slices of my moving 3D presence, changing and transmogrifying in bizarre, cartoony, sci fi fashion---appearing from nowhere of a sudden, and changing insanely, cross sections of my feet, legs, belly, head, then gone.
Poof.
What the hell was that?
***
The math profs will get mad if you say the 4th D is time.
But for our POPULAR understandings of all this, please, the 4th D is time.
The profs would exactly say: time is just one of an infinite number of dimensions beyond length, width and depth.
But, please, prof, it's hard enough as it is.
***
Einstein undoes Newton, the Physics Prophet of his millenium.
They both published each their 3 world changing discoveries at age 26.
Einsteins in the late 10's or early 20's.
He was confirmed in the 20's, I'm pretty sure, when the gravity of the sun was indeed found to BEND light rays from stars just behind our sun, from our vantage.
So close to the sun, those stars, they can only be seen during a total solar eclipse.
When it was found that the stars whose light rays, to reach us, must pass close by the sun, when it was CONFIRMED that those rays did indeed bend, well, Einstein was pretty much proved.
He's been confirmed in all ways since, sub atomic particles near the speed of light, the energy in the atom...
The Prof