"Why don't you use my old one that you provedas a kid?" advised Pythagoras. He goes to the blackboard andwrites:
(d)2=(x)2 +(y)2
Einstein says, "Yes, but I think I will use theone with the four dimensions plus a time dimension.
He goes to the blackboard and writes one moreterm:
(d)2=(x)2 +(y)2+(z)2+(vt)2
Einstein scratches his head and says, "If Ireplace dimensions x, y, z with symbols that mean momentum, mumble,mumble, mumble."
Einstein fills the blackboard with symbols,erases, writes again and finally steps back.
(E/c)2=(Mc)2 +(p1)2+(p2)2+(p3)2
He steps back and says, 'The p's are momentum:if the mass isn't moving we can make those zero and then reduce theequation to:
E=Mc2
Who will ever do anything with that?" hequestions.
The commentator returns and says, "That's theway Physics and Mathematics are. People create or discover thingsthat may not be of much use in their time. Sometimes much later itgets used. Remember, in 1905 when Einstein came up with this famousformula, most people rode in horse-drawn buggies and the airplanehad not been invented.
"Now, we will address another idea that was waybefore its time, higher dimensional spaces, specificallyMinkowski's eight–dimensional spaces. We will let Geroiamo help outin the explanation."
We see Professor Minkowski in his academicrobes speaking before a class, and drawing on ablackboard."
"Let me introduce you to an eight-dimensionalconcept of spacetime. The first four dimensions are those of commonexperience. We can have one dimension of front-back, one ofleft-right, one of up-down, and another for time that could bebefore-later.
"Let me illustrate with this three-dimensionalcheckerboard."
Professor Minkowski goes over to a structurethat is four 8 x 8 regular checkerboards, one above the other,separated by plastic legs.
He places a black checker piece at the cornerof the bottom board and says,
"Here, we have a three dimensional space. Fornow, we will let velocity equal zero. The piece can move forward orback, let me call that x,right or left, let me call that y, or up or down, let me call thatz. Let's call this corner the zeroof all dimensions. Now, I will move this piece up four, to the toplayer, forward four spaces, and left four spaces. It is now at z=4,y=4, x=4. How do we figure out how far the piece has moved from thezero corner?"
Pythagoras appears in his toga from the side ofthe stage. He says," All you have to do is use my theorem." Hewrites on the blackboard:
(d)2=(x)2 +(y)2+(z)2 +(vt)2=16+16+16+0=48
Minkowski produces a calculator from his pocketand says,"d equals the squareroot of 48 that is 6.93. If we had moved the whole checkerboardstructure through the time dimension such that vt=1, then d wouldbe 7 even."
"In these four dimensions, shown here, all ofwhat you might think of as normal physics taught in our k-12schools applies."
Minkowski then places another set of fourcheckerboards, made out of clear plastic, on top of the other four,with the zero corner located where the piece is at 4,4,4. He says,"Here, we are adding four mote dimensions that start from where thepiece is in after moving in the first four. Since we used ournormal four dimensions in the bottom checkerboards, we have to useimaginary numbers here."
Geroiamo walks in from the side of the stageand says, "Don't worry about the idea of imaginary numbers: theyare simply another kind of numbers that are convenient formathematicians."
Minkowsky continues, "The physical piece, hereat location x=4, y=4, z=4, can't move into the imaginary space.That is the law of physics. However, information about the piececan move into all eight dimensions. It can be up here in theimaginary space at ix=4, iy=4, iz=4."
The commentator returns to say, "Here, we haveto say that this is a new theory. To prove Pythagoras' theorem, allwe have to do is go out and measure a bunch of triangles and seewhether it worked. We know that Einstein'sE=Mc2 idea behind all ournuclear power plants. Later, we will show you examples of howinformation travels in eight–dimensional space."
Minkowski returns and continues,
"If our physical piece is at location x=4, y=4,z=4, and has eight-dimensional information about the piece atcoordinates of the x=4, y=4, z=4, ix=4, iy=4, iz=4 and t=it=0 (sowe don't have to bother with time here) the information then, wecan calculate the information distance between the zero corner (onthe bottom checkerboard)."
Then, Pythagoras reappears and says, "We canuse the eight-dimensional form of my formula." He writes hisformula mumbling to himself:
(d)2=(x)2 +(y)2+(z)2 +(vt)2+(ix)2 +(iy)2+(iz)2 +(ivt)2
He says, "Since we are letting t=0.we canrewrite this as:
(d)2=(4)2 +(4)2+(4)2 +(0)2+(i4)2 +(i4)2+(i4)2 +(0)2 "
Geroiamo jumps up and says, "Since(i)2=-1,
(d)2= 16 +16+ 16 −16 −16 −16 = 0 "
Minkowski returns and says, "Theeight-dimensional information distance between the starting squareon bottom checkerboard and the physical piece is zero."
Einstein returns and says," When I firstlearned about all of this I was a little tyke, when the wordsPythagoras and hypotenuse were beyond me. My uncle, Herman wholived a few miles away, explained it this way:
To get to my house from your house, you have togo down the highway for four miles, turn left on the crossroad andgo three miles, and there you are. Or, you could not go by the roadand take the shortcut across the field directly from your house tomy house. You calculate how long the shortcut is by squaring thedistances on the two roads adding them up and taking the squareroot. 42+ 32= 25=52. The shortcut to my house is 5miles.
I like the word shortcut better thanhypotenuse."
Minkowski says, "I agree, lets not confusepeople. Let's call the distance the shortcut distance."
The commentator returns and says, "What doesall this complicated mathematics mean? It means that, ineight-dimensions there is a zero-distance information shortcut fromthe corner square to where the checker started to where it is onthe fourth level. If you were at the starting square and wanted toknow some information about the physical piece (if it is heads-upor heads-down) The information could come through theshortcut.
"You don't need to care about or understand allthese mathematics. You do need to know that a valid, scientific,paradigm exists for the many kinds of information shortcuts we useand observe.
"We all have something I call the 'Magic Mirrorof The Mind.' In fairy tales, some witches, or sorcerers have magicmirrors that they can command
