Koch Island

To play with this image - tesselate it, check out various stages of the fractal and so on, go to http://wchow.home.znet.com/koch.htm

Dwell in Possibility

The following is taken from the site "What is Four-Dimensional Space Like?"

If you were to live in a tesseract, you might choose to live in its three dimensional surface, much as a two dimensional person might choose live in the 6 square rooms that form the two dimensional surface of a cube. So your house would be the eight cubes that form the surface of the tesseract. Imagine that there are doors where ever two of these cubes meet. If you are in one of these rooms, how many doors would you see? What would the next room look like if you passed through one of the doors? How many doors must you pass through to get to the farthest room? How many paths lead to that farthest room? Could you have any windows to outside the tesseract? What about windows to inside the tesseract?

Some of these questions are not easy. To answer them, go back to the easy case of a three dimensional cube with faces consisting of squares. Ask the analogous questions there and just extrapolate the answers to the tesseract.

The Fourth Dimension, Time, and Plato's Cave

This video comes from Rob Bryanton's Blog, Imagining the Tenth Dimension. If you are interested in higher dimensions you might want to check out his post that explores a specific asepct considered in 0, 1, 2, 3, 4 and all the way up to 10 dimensions.

Math and Mind Reading

Can YOU figure out HOW this works?

I have this one posted on my "Fun Math" page too - just the video. I haven't had a chance yet to think about the math behind it, although I'm certain it is based on mathematics. So I thought I'd post it here as well, under "explorations" to see if anyone wanted to comment on how this works - and beat me to it! Have fun!


The Prediction @ Yahoo! Video

Moby Dick and Princess Diana?

from: http://cs.anu.edu.au/~bdm/dilugim/diana.html copyrighted by Brendan McKay (bdm@cs.anu.edu.au)

WOW!! Herman Melville predicted the death of Princess Diana! If you click on the image above to enlarge it you will see very clear patterns with the words "Diana," "Dodi," "skid," "hearse," "royal," "Lady Diana," "mortal in the jaws of death," and even "Henri Paul!" BUT THAT'S NOT ALL!! We can also find in Moby Dick predictions of the assassinations of Indira Ghandi, Sirhan Sirhan, Abraham Lincoln, John F. Kennedy and many others!! (click here) Melville was an amazing prophet!

OR WAS HE?

Actually, the page at which I found the above image and information was posted in response to a challenge put out by Michael Drosnin who claims that codes in the Bible (found in patterns of letters in the Hebrew) have predicted past and do predict future assassinations. It goes without saying that there are critics of his work. In an interview for Newsweek Magazine, Drosnin said he would believe his critics if they could find a message about the assassination of a prime minister in Moby Dick. Well...they did...and then some!!

You see, given a sufficiently large collection of letters or numbers or points or events or people, there will ALWAYS be patterns (highly organized subsystems). In a large collection (like the letters in a novel) you will always find "hidden" words and patterns. The question is, are they MEANINGFUL? The answer is, "No." This is just a natural consequence of having a large group of items. Just because the patterns are not meaningful, does that mean it's not fun to play around with them - NO! It can lead to really cool, fun things such as the intriguing and adventerous National Treasure movies. This is all great fun, as long as people don't take it seriously!

As mathematician T. S. Motzkin says:

Complete disorder is impossible.

If you're interested in looking into the math behind this, it is part of the branch of mathematics known as Ramsey Theory.

Point-9 Repeating and 1

Well, this is one way to use the retired number 8 on a race-car, but only if you believe that point-9 repeating is exactly equal to one, making 7.9 (where the 9 repeats) exactly equal to 8.

I just finished putting up a video clip about this on my tutorial site (Meyer Math), but then I came across a great page that spells it out using even more proofs than I did and also includes a section of arguments and responses.

Largest Prime & Number Names

The following image is from the page Known Mersenne Primes . Notice that if you go to that page you have the option to click on the word name of large primes. Do you know your place value beyond ones, tens, hundreds, millions, billions, trillions, quadrillions, quintillions?
How about:

quattuormilliamilliatrecensexviginmilliaunsexagintillion

Were the largest currently known prime (the one referred to in the bottom row of the image above) to be printed out it would BEGIN with:

316,470,269,330,255,923,143,453,723,949,337,516,054,106,188,475,264,644,140,304,
176,732,811,247,493,069,368,692,043,185,121,611,837,856,726,816,539,985,465,097,356,
123,432,645,179,673,853,590,577,238,179,357,900,876,426,103,943,782,376,494,591,742,
934,588,497,117,587,146,916,972,984,761,159,060,873,250,939,462,085,575,740,754,577,
098,620,558,011,779,529,884,042,198,287,643,319,330,465,064,455,234,988,142,139,565,
785,447,474,023,546,353,758,537,324,801,838,120,387,600,868,416,525,400,790,381,285,
888,256,687,085,855,456,231,577,527,939,305,920,811,766,585,308,670,132 . . .

. . . and would end with . . .

984,625,380,787,247,802,532,758,511,333,502,460,778,884,339,034,019,700,927,663,958,
167,698,908,010,736,101,410,136,996,852,925,703,272,553,544,622,464,685,928,707,526,
568,105,993,689,915,218,073,801,443,404,945,008,266,425,932,413,139,826,915,084,069,
991,159,279,791,908,398,130,223,304,824,083,119,093,195,998,014,562,456,347,941,202,
195,900,928,079,670,729,447,921,616,491,887,478,265,780,022,181,166,697,152,511

. . . with THOUSANDS of pages in between to write out the complete number. We are always on the lookout for new primes. We can prove that there are infinitely many primes - that we will never run out. This was proven by Euclid a LONG time ago, but large primes are very hard to find. If you want to join the hunt (with a possibility of big money for YOU while your computer does all the work), go to the Great International Mersenne Prime Search. The information on this page was taken from the Prime Page of Landon Curt Noll, pictured at right.

Mandelbrot Set (Basics)

Explanation to come (post under construction).

First image from Wikipedia, permission granted to copy.
Second image from: H. Jurgens, H.O. Peitgen, and D. Saupe ©copyright 'The Beauty of Fractals' by H.O. Peitgen, P. Richter (Springer -Verlag, Heidelberg, 1986) and 'The Science of Fractal Images' by H.O. Peitgen, D. Saupe (eds.0 (Springer-Verlag, New York, 1988)

Hilbert Curve Fractal

The Hilbert Curve is an example of a (linear) fractal. In particular it is a plane-filling curve, the first example of which was discovered by Giuseppe Peano in 1890. Plane-filling curve is an oxymoron along the lines of "pretty ugly" and "jumbo shrimp." This is because curves are one-dimensional, but planes are two-dimensional, so to have a curve that can completely fill part of a plane (i.e. become two-dimensional and yet remain a curve) is somewhat of a contradiction in terms.

It is called a linear fractal because at its final stage any small portion of it can be magified to become an exact copy of the whole. Fractals are created through a process of iteration, following a rule over and over. The first three stages of the creation of this curve are shown in the diagram above. The square is cut into 4 pieces, then 14, then 64. The curve is a path through the square that hits the middle point of each of the sub-divisions of the square. There is a specific process for the creation of this curve. The shape in the previous stage is copied 4 times, shrunk by a factor of 2, copied 4 times over - twice in the bottom with the same orientation as before and twice on the top, once rotated counter-clockwise and once rotated clockwise. The 4 copies are then connected by a line segment (shown above by dotted lines).

It's much easier to see than to say!!

These images were taken from the Plane Filling Curves page at the wonderful Cut the Knot site, which contains "interactive mathematics miscellany and puzzles." Speaking of INTERACTIVE, if you to the Plane Filling Curves page you can use a Java Applet to click and see the stages go from one to the other.

Hypercube Slices

The image below is a still screenshot of an interactive hypercube-slice applet found at the site Projections of Hypercube Slices by Davide P. Cervone. It is one of the links from his phenomenal page Some Notes on the Fourth Dimension. This site has many other interactive links.When you go to this site, play with the controls you see here under the applet for some cool effects!

As in the wonderful work of fiction Flatland by Edwin A. Abbot, objects from higher dimensions are often studied by imaging slices of those objects as they pass through a lower dimension. Another site that contains applets showing animated 3D slices of a hypercube is Alice begegnet der vierten Dimension . I like the continuous motion of the animations. The longer you watch the better feel you can get for what is going on.


Below are still shots of a hypercube going through 3-dimensional space. These are taken from the site HyperSpace, User Manual by Paul Bourke. Though it is possible to comprehend each solid shown here one at a time, in order to grasp the full 4-dimensional object, you need to keep all of them in mind at one time and "stack them up," just as someone in a 2-dimensional world would have to "stack up" a series of smaller to larger to smaller circles to grasp the idea of a sphere.

To help you get an idea of what is going on here I created a series of slices of a sphere going through "Flatland." (Imagine a ball being submerged in water and the part of it that is at water level as it passes through the surface of the water.) In order to understand what a sphere is, a thing that cannot exist inside of Flatland, the Flatlanders would have to keep all of these slices in mind and stack them up.

Powers of Ten

At Secrets Worlds: The Universe Within there is a great Java Applet that zooms in and out from 10 million light years from earth to the level of quarks inside an atom on earth. You can have it run automatically both in and out, or you can manually click from one level of magnification to the next.

It's fun to watch and also a GREAT way to get an idea of scientific notation, exponents, and powers of ten.

Image above by Chuck Higgins (at www.higginsc.com/Deepsky/deepsky.htm)

Fractal Zoom

This is one of my favorite zooms into the Mandelbrot Set of those available on YouTube.

Number Factoring Machine

Do you have a big number you need to factor (or a big number that you need to determine whether it is prime or composite)? Thanks to the guy pictured on the left, and his programming skills, you can just type your number into the space provided below, and poof!, your work is done for you!

This solver is provided by Algebra.Com:

Rotating Hypercube

The object you see is the four-dimensional analog of a cube, just like a cube is a three-dimensional analog of a square. As you watch this video, realize there are no cubes on the "inside." The cubes you see are the outside of the shape just as squares are the outside of a cube. Look at the perspective drawings of a cube below. In the one on the right the smaller square is NOT "inside" the cube. It's just a matter of perspective, as if you are looking into a box, and the back is farther away than the front. Same thing with this video. The hypercube is ROTATING. It is NOT turning itself inside out.