Educator How-To: Nautilus and the Golden Spiral (an approximation)

Editor’s note: In honor of our new Nautilus Live program — which takes Museum patrons to the ocean floor with telepresence technology — this month’s Educator How-To is all about the nautilus shell. From our veteran Xplorations educator Kat Havens:

It is difficult to deny the beauty and perfection of the nautilus’ spiraled chambers. Many have heard it is a perfect example of a Golden Spiral or have seen pictures of it neatly fitted into a Golden Rectangle. Although compelling, it is mathematical mythology.

Untitled

The angles found within the chambers of the shell exhibit multiple angles that are not congruent with those of the Golden Spiral. In fact, the spiral of the nautilus is more correctly known as an approximate logarithmic spiral or as exhibiting logarithmic spiral growth. Growing in this manner allows the animal to increase in size without changing its shape. We think that it is an excellent example of Mother Nature’s knack for beautiful symmetry.

Materials:

•    Cut nautilus shell
•    Styrofoam or paper plate
•    Sheet of craft foam or other padded surface
•    Paper – color of your choice
•    Acrylic paint – color of your choice

paint print

Procedure:

1.    Gather all of your supplies. Cut nautilus shells are common and may be purchased at seaside shell shops or found online at a reasonable price. They are reusable, provided they are cleaned promptly after each use. We tried cutting our own shells with a fine saw but, we were not completely satisfied with the results.
2.    Place “springy” material, such as craft foam, under the print paper. This allows for a better “pull” as the give in the foam allows for better contact between shell and paper.
3.    Pour a good amount of acrylic paint onto the plate. Manipulate the plate by tipping it around until the paint is spread in an even layer that is large enough to accommodate the shell. Place the shell into the paint and pull it out. You will find the paint may coagulate in the smaller chambers and makes an unclear print. This is solved by gently blowing on these chambers to break the paint bubbles.
4.    Carefully press the shell onto the paper.  Do not move the shell in any direction once contact with the paper is made as it will smear the print. Gently pull the shell up. Often, you can get another decent pull directly after the first print, so feel free to make two or more with one paint application.

nautilus print

Happy Birthday, Isaac Newton!

Had he lived to see it, Sir Isaac would be 367 years old today – and probably pretty amazed at the scientific leaps and bounds we’ve seen since the 1687 publication of his Principia – widely regarded as one of the most influential books in the history of science.

Though he’s known to schoolchildren to world over as the recipient of a nasty bump on the head from a falling apple – the true origin of Newton’s conceptualization of gravity comes from a little higher in the sky. So, in honor of Sir Newton’s birthday, here’s a short clip from the BBC explaining how we came to know why we don’t just fall right off the Earth:

How do you remember Newton? Let us know in the comments.

And if you haven’t already, check out Google’s homepage today for their celebration (be sure to scroll over the image to get the full effect).

UPDATE: According to Scientific American, the apple story is not as apocryphal as some have claimed. (via BoingBoing)

Can you conquer the Tower of Hanoi?

Puzzles are very good at making you think flexibly and enabling you to find patterns (skills great for science and pretty much everything else), but they’re also just fun. One classic logic puzzle is the Tower of Hanoi, invented by Edouard Lucas in 1883.  This puzzle has been one of my favorites since I first saw it about twenty years ago.  As with many problems, there are multiple ways to achieve the basic goal, but after exploring you may figure out the most elegant or efficient way to get there.

One seven-disk version of the Tower of Hanoi looks like this (many more can be seen here):

The goal is to get the stack from the left pin to the right pin, finishing with the stack in the same order (largest disk on the bottom, smallest on top, and all the other disks in order). 

The rules:

1) You may only move one disk at a time. This means you may only move the topmost disk in a stack.

2) You may move the disk to any of the three pins, HOWEVER:

3) You may never stack a larger disk on top of a smaller one.

It’s easy to get the idea with just a few disks:

Here is the starting position:

Then we move the small disk to the middle pin:

Which allows the big disk to go to the last pin:

Then the small disk stacks on top of the big one and the stack is complete:

Try making and moving a three disk stack (just remember you can’t stack the larger ones on the smaller ones!):      

You can also use a dime, nickel and quarter, or three other coins of differing sizes, and just mark ‘tower slots’ on an index card in place of the pins.

 .

Once you get the three disk stack moved successfully, see if you can do it in only seven moves (two disks took three moves).  Four disk towers can be moved in a minimum of 15 moves.  You can also try this online. This version lets you choose up to eight disks and will keep track of how many moves you make.

Here are some things to think about as you play:

Do you start to see patterns in the way you move the disks?

Is there a rule about where you should put the first disk when moving a stack of three, four, five, etc.? Does it matter if the number of disks is even or odd? Does the pattern continue no matter how many disks you have in the tower?

What is the minimum number of moves for five disks?  Does this fit into any kind of pattern with the previous ‘minimum move’ numbers?   The minimum number of moves for a seven-disk stack is 127; does this fit the pattern?

One apocryphal story tells of a tower of 64 golden disks, which when completely moved, signal the end of the world.  Assuming one move every second, this stack would take over 500 billion years to move.

Check out this site for more discussion of the Tower of Hanoi puzzle.