Max disovers pi , but still has to learn that it is a Greek letter that looks like a double T, and that it stands for the number 3.1416 . . .

For more of Max's discoveries, go to the next page of
Miniaturists' Mathematics
To work in one-twelfth scale means that we are dividing by twelve.  Dividing what?  That's a good question.  In this picture you can see twelve little people who have obligingly arranged themselves to show that each of them is one-twelfth the
height
of the big man beside them.   
But as you can tell from the spectacles and the ruler, this man in th e blue shirt is to human eyes only six inches tall. .   
Here 12 blue-shirted men are standing on one another's  heads to make up the height of the toymaker.  

The  blue-shirted men are 1/12th the height of the toymaker, and the little people in the top pictures are then 1/12 of 1/12, or 1/144 the height of the toymaker.  If I had 144 of them standing on one another's heads, the top ones would be on a level with a human head.

Applicable mathematical rules:

!. "of"  means "multtiply."  For instance, 1/12 of 1/12 equals 1/144.

2.  To multiply fractions, multiply the two upper numbers (1 x 1 = 1) and the two lower numbers (12 x 12 = 144).

3.  If you want to go a further step and "divide" the smallest people into midgets 1/12 of their height, you would need to pile up 1728 dolls to reach human height.  1/12 x 1/12 x 1/12 =1/1728.  The small picture shows one lady with her height divided by 12.  The tiny triangle on the right of her is the size of doll we'd be talking about.
See point 3 below
Mild mathematics
1/12 and 1/144
Frances Armstrong
Circular secrets
If you see numbers as your enemy, you will need to find some special friends, and a pair of compasses is (or are) friends worth making.  The one in the picture below came from the dollar store, and looks better than it behaves.  Floppy compasses are a menace, as  you can see from the uneven circles this one has drawn.  (Or I could blame the little chap in the picture.  His name is Max.)
You probably already know how to draw this simple but impressive design, but if your education was totally math-free, you can learn it now.    And several other lessons at the same time.
Once you have obtained your compasses, you can begin.  You don't even need squared paper--anything will do.  

This page has turned out dark, perhaps because I headed this section "Circular Secrets."  But you should be able to see three circles.  Start with the one at top left. Draw a circle on your own paper, any size.

Now keep the compasses set as they were, so that if you drew another circle it would come out the same size.  Choose any point on the circle's circumference (the part that you have drawn), and

make a little mark.  Then put the pointed end of the compasses on the mark, and swing the pencil end round till it meets the circle again.  Make another mark, and repeat the process. Unless you've been dreadfully careless, you will happily find yourself ending up where you began.  You now have six little marks equally spaced around the circle. 

You can do various things now. 

1.  Join the dots (or more accurately, the places where lines crossed) in the obvious way , and you will have a nice hexagon, a six-sided figure (bottom left).

2.  Alternatively, you can make the flower design by using the same procedure as when  you made the marks, only this time letting the pencil point stay on the paper while it travels within the circle.  Move the compasses to the next point where pencil lines have crossed, and make another partial circle (arc). 
It's sometimes surprising how very small even a 1/12 object can be.  Take a footprint--any size--and then try to draw one  twelve times as big, and one twelve times as small.  You'll soon run out of paper or screen.
5.  You  now have six identical triangles.  They are identical in two senses: they look the same as one another, and each of them has three identical sides.  (Terminology: triangles that look the same as one another with angles and sides matching, are called  congruent triangles.    Triangles which have three identical sides (and three identical angles) are called equilateral triangles.

You can find simpler ways of drawing such triangles, and may discover other interesting things, like how to divide a line in half.  I'll come back to some of these later.



3.  From here you can go on to make many more patterns: for instance, try joining dots in alternate pairs: first corner of hexagon joins to third corner, second to fourth etc. You should get a kind of star effect.
4.  Next, go back to the hexagon (picture above, bottom left) and join each of the six points to the centre of the circle.  (It will be easy to find the centre, because the point of your compasses will have made a hole there; but you can probably find a way using your compasses only.  I'll come back to this another time. ) 
Little Learning    Littleness and miniaturization     Language and littleness       New Projects
">
">
">
">
6. One last note about circles for now: someone asked how to find the measurement around a circle, if you know the diameter of the circle.  This is something you will need to do if you are trimming a circular tablecloth.  There is a simple formula, circumference equals diameter multiplied by pi, pi being the name of a letter of the Greek alphabet.  It stands for the number 3.1416 . . . -- the decimals go on for ever).  This bit of arithmetic will work for any kind of circle.

If you don't like the sound of 3.1416, try this.  Go back to my last diagram, and notice that the circumference is broken up into six curves (arcs).  Notice also that each section of the circumference is paired with a straight line (see inside orange border).  Now there are six lines like that, and each one is the same length as the radius. 


Those six straight lines, added together, are quite close to being the circumference of the table.  You can see this, because each of the six straight lines is nearly the same as its curved  partner. So if we added up the length of those lines, we will get a number that's not too far wrong.

And we do know the length of the straight lines I'm talking about: they are each half the size of the diameter. 

So the circumference is, approximately, six lots of lines the length of the radius, which means three times the diameter.  And this is close to pi (3.1416) times the diameter.

Better to use pi if you want to be sure your braid or fringe will go all the way round the edge of  your tablecloth.

Max disovers pi , but still has to learn that it is a Greek letter that looks like a double T, and that it stands for the number 3.1416 . . .

For more of Max's discoveries, go to the next page of
Miniaturists' Mathematics
">
">
">
">
">
">
">
">