Extra-dimensional theories: making the jump to hyperspace
Serious people with serious titles at serious institutions
would have us believe that the universe probably has more
spatial dimensions than the three we can see. The more I consider
this improbable-seeming idea, the more convincing I find it.
Much of what follows is inspired and informed by the writings
of Michio Kaku, Lisa
Randall and Brian
Greene.
Try this fun
interactive simulation of a hypercube, a four-dimensional
cube. You can turn it around with the mouse and see the
different 2-D 'shadows' it casts on the plane of the computer
screen. See other hypershapes:
a hyperstar,
a pentatope,
a knotted
torus. See
a demo for a four-dimensional video game! I've seen people
call Katamari Damacy
a four-dimensional game as well.
Let's say we woke up one morning able to see and move along
the fifth dimension (the fourth spatial dimension.) It would
be like always being able to move at a right angle to reality,
as Douglas Adams put its. If you picked up a right shoe, turned
it around a hundred eighty degrees along this new axis and
set it down, it would be transformed into a right shoe. Another
way to think about this: turning something a hundred and eighty
degrees in the fifth dimension is same as turning it inside-out.
When you're looking at yourself in the mirror, you're not
seeing yourself swapped left-right (why then aren't you upside
down?) You're seeing your eversion, how you'd look rotated
in the fourth spatial dimension.
Speaking of Douglas Adams. In So
Long And Thanks For All The Fish, the fourth book (of
five)
in the Hitchhiker's Guide trilogy, Arthur and Fenchurch arrive
at the home of Wonko The Sane.
What it was like was this:
It was inside out.
Actually inside out, to the extent that they had to park
on the carpet.
All along what one would normally call the outer wall,
which was decorated in a tasteful interior-designed pink,
were bookshelves, also a couple of those odd three-legged
tables with semi-circular tops which stand in such a way
as to suggest that someone just dropped the wall straight
through them, and pictures which were clearly designed to
soothe.
Where it got really odd was the roof.
It folded back on itself like something that MC Escher,
had he been given to hard nights on the town, which is no
part of this narrative's purpose to suggest was the case,
though it is sometimes hard, looking at his pictures, particularly
the
one with the awkward steps, not to wonder, might have
dreamed up after having been on one, for the little chandeliers
which should have been hanging inside were on the outside
pointing up.
Confusing.
Fortunately, with some practice, visualizing the higher dimensions
is actually kind of fun.
Consider the Necker cube
Is this image two-dimensional or three? Most of us would
answer three. Some might say, more accurately, that it's the
two-dimensional shadow cast by an imaginary three-dimensional
object. It's easy for us to mentally reconstruct the 3D original
because we've encountered cubes in real life. But it's important
to remember that outside of your imagination, there's nothing
3D about the Necker cube. Note also that in the absence of
shading or perspective, your brain can't quite decide which
face is the 'front'.
It's harder to extrapolate from a 2D shadow to a 4D or 5D
or 6D object, but the principle is the same.
Consider nonstick frying pans
Excerpts of an article by postdoc student Claire Irving in
The Wrangler,
the University of Leicester’s math department newsletter:
You may wonder what wallpaper and frying pans have to
do with each other.
In everyday life we see geometric patterns all around us.
Decorative items, such as wallpaper or tiles include patterns
in their designs. Patterns also appear in nature, in the
arrangements of leaves on a plant or petals on flowers.
Also, animals often have patterns in the colorings on their
fur, feathers or skin. The patterns described above contain
symmetry. Symmetries can be reflections, such as is seen
in the wings of a butterfly, rotations, seen in the arrangement
of petals on a flower, translations, such as in snakeskin,
or glide reflections, combining reflection with translation,
which can also be seen in snakeskin.
Patterns involving translational symmetry are called periodic.
Motifs in periodic patterns can be shifted on to other motifs
so that every one matches up with another. To see this,
you could copy the pictures in this article on to a transparency
and then move the copy over the pictures to find when motifs
coincide. Patterns with translational symmetry in one direction
are called friezes. By considering the different symmetries
which friezes can have, it can be shown that there are only
seven distinct types. Patterns with translational symmetry
in two directions are called wallpaper patterns. The symmetries
of these patterns have also been studied, and it is well
known that there are seventeen different types of wallpaper
pattern.
Patterns with translational symmetry in three directions
are called crystals. An example is the atomic lattice for
diamond. A more unusual kind of pattern is quasiperiodic.
Like wallpaper patterns, there are motifs which repeat,
but the patterns do not repeat in a periodic way. If we
shift one of the motifs on to the other one, the shapes
do not match up.
There are also three-dimensional structures which are quasiperiodic.
Such objects are called quasicrystals. For example, certain
alloys of aluminum, copper and iron have quasiperiodic atomic
arrangements. The fact that there is no translational symmetry
in their atomic arrangements, and hence no fault lines in
their structure, means that quasicrystals are often quite
hard. Also, their unusual structure means that other substances
cannot find ways to grip on to quasicrystals, so they have
non-stick properties. This has led to their use as coatings
for frying pans.
A Penrose
tiling is an aperiodic tiling of the plane discovered
by Roger Penrose in 1973. Here's an example:
Being aperiodic, it has no translational symmetry - it never
repeats itself exactly, but nevertheless it has a fivefold
rotational symmetry. Lisa
Randall says that a quasicrystal is the three-dimensional
shadow cast by a higher-dimensional object, and that quasicrystals
become periodic when viewed in enough additional dimensions.
Consider the complex numbers
In mathematics, a complex number is a number of the form
a + (bi), where i2 is -1. Except
the weird thing here is that any positive number times itself
is positive, and any negative number times itself is also
positive, and zero times itself is zero, so what the heck
kind of number is i? Imaginary! For example, 3 +
2i is a complex number, with real part 3 and imaginary
part 2. Real numbers may be considered to be complex numbers
with imaginary part set to zero; that is, the real number
a is equivalent to the complex number a+0i.
The words 'real' and 'imaginary' were meaningful when complex
numbers were used mainly as an aid in manipulating 'real'
numbers, with only the 'real' part directly describing the
world. Later applications, and especially the discovery of
quantum mechanics, showed that nature has no preference for
'real' numbers and its most real descriptions often require
complex numbers, the 'imaginary' part being just as physical
as the 'real' part.
In electrical engineering, you can unify the mathematical
modeling of resistors, capacitors, and inductors by combining
all three in a single complex number called the impedance.
This use is also extended into digital signal processing and
digital image processing. In fluid dynamics, complex functions
are used to describe potential flow in 2d. Certain fractals
are plotted in the complex plane, for example the
Mandelbrot set and Julia set. What follows are excerpts
from a wonderfully
lucid web site run by Prof Philip Spencer of the University
of Toronto.
It may seem hard to believe that imaginary numbers could
possibly exist. The source of this difficulty stems from
what one means by 'existence'. In mathematics, whether or
not a certain concept exists can depend on the context in
which you ask the question.
When talking about numbers, there are many very different
contexts that one could have in mind. Here are the
four most familiar ones:
- The Natural Numbers. These are the counting numbers
1, 2, 3, etc that are possible answers to the question
"how many?" They are abstract concepts that
describe sizes of sets.
- The Integers. These are abstract concepts that describe,
not sizes of sets, but the relative sizes of two sets.
They are the possible answers to the question "how
many more does A have than B has?" They include both
positive numbers (meaning A has more than B) and negative
numbers (meaning B has more than A).
- The Rational Numbers. These are abstract concepts that
describe ratios of sizes of sets. They do not model sizes
of sets the way that natural numbers do. If you say "I
ate 3/4 of a pie", you are not saying that the set
of things you ate had 3/4 elements. Instead, you are expressing
a ratio of two integer quantities: 3, the number of pie-quarters
that you ate, and 4, the number of pie-quarters that make
up a whole pie.
- The Real Numbers. These are abstract concepts that
describe measurements of continuous quantities, such as
length, weight, quantity of fluid, etc. (Don't let the
word 'real' fool you; the real numbers are no more 'real'
in the ordinary English sense of the word than are any
other kind of numbers.)
Concepts that exist in one of these contexts may not exist
in another. The question "does there exist a number
between 1 and 2?" has the answer no in the first two
contexts (you cannot go to the beach and pick up more than
one but fewer than two pebbles), but yes in the last two
contexts (you could eat three cookie halves, which is in
between one whole cookie and two whole cookies).
Although in the first two contexts there does not exist
a number between 1 and 2, most people are quite comfortable
with the fact that such numbers do exist in other contexts.
For instance, people don't usually have trouble accepting
the existence of the fraction 3/2. Why then is it so hard
to believe that the concept of 'a number whose square is
-1', though it does not exist in any of the four contexts
mentioned above, might nonetheless exist in some other context?
It is because we usually forget the fact that we already
have four quite different meanings for the word 'number'.
We have become so familiar with each of the four contexts
that we have jumbled them together in our mind as if they
were a single concept. When we encounter a notion like 'square
root of -1' which does not exist in any of these four contexts,
we think that it cannot exist at all, because we think the
word 'number' is a single concept that embodies just these
four contexts.
Instead, what we should be thinking is something like this:
- Okay, I know about four different number systems: one
in which 'number' means a measurement of how many items
are in a set, a second one in which "number"
means a relative measurement of the sizes of two sets,
a third one in which 'number' means a ratio of sizes of
two sets, and a fourth one in which 'number' means a measurement
of a continuous quantity.
- In neither of these four number systems does there
exist a square root of -1.
- Might there be a fifth context, a number system (where
'number' means something different from any of the above
four things) in which there does exist a square root of
-1?
The answer to that final question is yes. It is called
the Complex Number System. Although it will involve a notion
of 'number' that is something different from what we are
used to, the difference is not fundamentally any greater
than is the difference between the concepts of 'number of
elements in a set' (natural number) and 'ratio of sizes
of two sets'. In other words, the complex numbers are not
that much more different from familiar numbers than rational
numbers (fractions) are from natural numbers.
Okay, now we've seen that imaginary numbers exist. However,
they exist in the context of a different number system,
something different from the number systems we are used
to. The 'complex numbers' that make up this system are pairs
of numbers; do they really deserve to be called 'numbers'
in their own right? Well, remember that fractions are pairs
of numbers also. They clearly deserve to be called numbers
in their own right, since they can measure 'how much' in
some contexts (for instance, "I ate three quarters
of a pie"). So, the principle of considering a pair
of numbers (in this example, 3 and 4) as a number in its
own right is well established.
An imaginary number could not be used as a measurement
of how much water is in a bottle, or how far a kite has
travelled, or how many fingers one has. Nonetheless, there
are a few real world quantities for which complex numbers
are the natural model. The strength of an electromagnetic
field is one example. The field has both an electric and
a magnetic component, so it takes a pair of real numbers
(one for the intensity of the electric field, one for the
intensity of the magnetic field) to describe the field strength.
This pair of real numbers can be thought of as a complex
number, and it turns out that the strange rule of multiplication
of complex numbers has relevance to the physics of an electromagnetic
field.
Although such direct applications of complex numbers to
the real world are few, their indirect applications are
many. Many properties related to real numbers only become
clear when the real numbers are thought of as sitting inside
the complex number system. Therefore, complex numbers aid
in the understanding even of things that are described by
ordinary, familiar real numbers.
It's like trying to understand a shadow. The shadow lives
in a two-dimensional world, so only two-dimensional concepts
are directly applicable to it. However, thinking of the
three-dimensional object casting the shadow can aid in understanding
it, even though three-dimensional concepts don't have any
direct application to the two-dimensional world of the shadow.
Likewise, complex numbers may not be directly applicable
to a real world measurement any more than a three-dimensional
object is directly applicable to a two-dimensional shadow,
but they can still help us understand it.
Consider particle spin and handedness
If you rotate a left shoe one hundred eighty degrees in hyperspace,
you get a right shoe. The same is true of particles. If we
could see hyperspace - if the particles comprising both our
eyes and the light they're seeing weren't confined to our
three spatial dimensions - a pair of shoes might look amusingly
like two copies of a single object, side by side but pointed
in opposite directions. An analogy for us in 3D land would
be a pair of identical golf balls facing opposite directions.
Every particle has a measurable property called spin. Electrons,
protons, entire atoms and so on all behave as if they're spinning
around extremely fast at all times. A particle is 'right-handed'
if the direction of its spin is the same as the direction
of its motion, and it's 'left-handed' if the directions of
its spin and motion are opposite. By convention for rotation,
a standard clock tossed with its face directed forwards is
left-handed.
The laws of nature were long thought to remain the same under
mirror reflection, the reversal of all spatial axes. The results
of any experiment viewed in a mirror were expected to be identical
to the results in a left-right-reversed duplicate of the experiment.
This 'mirror symmetry' was known to be respected by classical
gravitation and electromagnetism; it was assumed to be a universal
law. However, in the mid-fifties, Chen Ning Yang and Tsung-Dao
Lee discovered that the weak interaction might violate mirror
symmetry. Chien Shiung Wu and collaborators in 1957 discovered
that the weak interaction does in fact maximally violates
mirror symmetry, earning Yang and Lee the 1957 Nobel Prize
in Physics. (But not Wu, the only woman involved, as Lisa
Randall exasperatedly observes.) In most circumstances,
two left-handed quarks will interact more strongly than right-handed
or different-handed quarks. Experiments which show this effect
imply that the universe has an otherwise unexplained preference
for left-handedness.
Now we start moving into MC Escher territory. Some people
speculate that a hidden 'mirror sector' exists in the universe,
where mirror symmetry is violated in the opposite way, so
the weak force acts more strongly on right-handed quarks.
Maybe it's in the evil universe where everyone has a goatee!
Consider spirals,
barber poles and Shepard tones
Dig mathematician
and unicyclist Mark Newbold's Java applet demonstrating the
counter-rotating spirals illusion. Be advised that it
may freak you all the way out and back.
A Shepard tone, named after Roger Shepard, is a sound consisting
of a superposition of sine waves separated by octaves. This
creates the auditory illusion of a tone that continually ascends
or descends in pitch, yet which ultimately seems to get no
higher or lower. Some examples:
The breathtakingly weird circular xylophone at the Exploratorium
in San Francisco
Pink Floyd - Echoes, at the end
the music accompanying the never-ending staircase in Super
Mario 64
The visual equivalent would be
MC Escher's Ascending and Descending or a barber pole.
'Circular' processes like clocks and the musical circle of
fifths are actually flattened spirals,
ie a lower-dimensional projection of a higher-dimensional
shape. Clock hands might sweep out a spatial circle, returning
to the same 'place' on a regular basis, but four o'clock today
is not the same 'place' as four o'clock today or tomorrow.
Each 'C' on the piano sounds like the 'same' pitch, but while
their harmonic functions are identical, they all sit in different
octaves.
Consider recursion generally
Here's a song I first encountered on the classic album Muppet
Silly Songs called I'm My Own Grandpa, by Dwight Latham and
Moe Jaffe.
Many, many years ago when I was twenty-three, I was married
to a widow who was pretty as could be
This widow had a grown-up daughter who had hair of red;
my father fell in love with her and soon they, too, were
wed.
This made my dad my son-in-law and changed my very life,
for my daughter was my mother, 'cause she was my father's
wife.
To complicate the matter, even though it brought me joy,
I soon became the father of a bouncing baby boy.
My little baby then became a brother-in-law to dad, and
so became my uncle, though it made me very sad
For if he was my uncle, then that also made him brother
to the widow's grown-up daughter, who, of course, was my
step-mother.
My father's wife then had a son who kept them on the run,
and he became my grand-child, 'cause he was my daughter's
son.
My wife is now my mother's mother, and it makes me blue,
because, although she is my wife, she's my grandmother too.
If my wife is my grandmother, then I am her grandchild,
and every time I think of it, it nearly drives me wild
For now I have become the strangest case you ever saw; as
husband of my grandmother, I am my own grandpaw.
Chorus: I'm my own grandpaw, I'm my own grandpaw,
It sounds funny I know but it really is so, oh, I'm my own
grandpaw.
Some more self-referential humor from the Internet:
This gubblick contains many nonsklarkish English flutzpahs,
but the overall pluggandisp can be glorked from context.
-David Moser
You have of course, just begun the sentence that you have
just finished reading.
-Peter Brigham
If you think this sentence is confusing, then change one
pig.
-Uilliam Bricken Jr.
Although this sentence begins with the word 'because',
it is false.
-Douglas Hofstadter
From Elegancelessness
by Paul Niquette :
Ubiquity is everywhere.
Never say never!
I never repeat myself; I never repeat myself.
All generalities have exceptions, except this one.
What if there were no hypothetical questions?
Why not ask rhetorical questions?
Is there another word for 'synonym'?
What is the opposite word for 'antonym'?
There was a young girl from Japan,
Whose poetry never would scan.
When she was asked why,
She said with a sigh,
"It's because I always try to include as many words
in the last line as I can."
There was a young boy from China,
Whose poetry was really much finer.
His limericks tend
To come to an end
Suddenly.
There was a young girl from Peru,
Whose limericks end at line two.
There was a young man from Verdun.
This sentence refers to itself.
Plurals are plurals in this sentence. Singular is singular
in this sentence. Plural is singular in this sentence. Singulars
are plural in this sentence. Plurals is singular in this
sentence. Singulars is singular in this sentence.
This sentence is written in the passive voice. This sentence
was written in the past tense. This sentence is in the present
tense. This sentence will be finished not in the future
but now. This sentence would have been expressed in the
subjunctive mood.
This sentence advises the reader to never split an infinitive.
Not to split an infinitive or to not split an infinitive,
this sentence advises to never do both or always to do neither.
This is the sentence that a preposition appears at the end
of.
Recursion, in mathematics and computer science, is a method
of defining functions in which the function being defined
is applied within its own definition. The term is also used
more generally to describe a process of repeating objects
in a self-similar way. For instance, when the surfaces of
two mirrors are almost parallel with each other, the nested
images that occur are a form of recursion. You can get even
cooler recursion by pointing a video camera at a monitor showing
its own output. Gerald Edelman
thinks that if such reentrant loops are a fundamental component
of your waking consciousness.
To understand recursion, one must recognize the distinction
between a procedure and the running of a procedure. A procedure
is a set of steps that are to be taken based on a set of rules.
The running of a procedure involves actually following the
rules and performing the steps. An analogy might be that a
procedure is like a menu in that it is the possible steps,
while running a procedure is actually choosing the courses
for the meal from the menu.
A procedure is recursive if one of the steps that makes up
the procedure calls for a new running of the procedure. Therefore
a recursive four course meal would be a meal in which one
of the choices of appetizer, salad, entrée, or dessert
was an entire meal unto itself. So a recursive meal might
be potato skins, baby greens salad, chicken parmesan, and
for dessert, a four course meal, consisting of crab cakes,
Caesar salad, for an entrée, a four course meal, and
chocolate cake for dessert, and so on until each of the meals
within the meals is completed.
Use of recursion in an algorithm has both advantages and
disadvantages. The main advantage is usually simplicity. The
main disadvantage is often that the algorithm may require
large amounts of memory if the depth of the recursion is very
large. A reentrant subroutine in a computer program is one
that can be safely called recursively or from multiple processes.
Compare to Gerald Edelman's concept of reentry.
A strange loop arises when, by moving up or down through
a hierarchical system, you find yourself back where you started.
Think of video games like Pac-Man and Asteroids, where the
screen wraps around. Strange loops may involve self-reference
and paradox. The concept of a strange loop was proposed and
extensively discussed by Douglas Hofstadter in Gödel,
Escher, Bach, itself quite a strange loop.
Gödel's first incompleteness theorem states:
Any formal theory capable of expressing elementary arithmetic
cannot be both consistent and complete.
G's second incompleteness theorem causes math people even
more distress:
If an axiomatic system can be proven to be consistent and
complete from within itself, then it is inconsistent. Therefore,
in order to establish the consistency of a system S, one
needs to use some other more powerful system T, but a proof
in T is not completely convincing unless T's consistency
has already been established without using S.
There are some who hold that a statement that is unprovable
within a deductive system may be quite provable in a metalanguage.
And what cannot be proven in that metalanguage can likely
be proven in a meta-metalanguage, recursively, ad infinitum.
Is the property of being self-unprovable itself self-unprovable?
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