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.