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## I Love Math

I love math, and I am not a mathematician.  I am a girl.  I am a girl who studied humanities and arts. I work in a field, and I have almost always worked in fields that require very very little math besides basic arithmetic come payroll time.  This blog is a cultural, intellectual, foodie, drinky, sort of blog.  But I’m blogging about math because I love it.  I hate that math is seen as geeky and useless.  There are so many beautiful patterns – things I cannot understand but can still recognize as stunning – and things that we as human beings still cannot understand.  Love this stuff.  I need to find an intro to number theory course – not only would it be fun, but it would keep me from doing stupid things like believing I discovered a pattern to predict prime numbers.  I won’t keep rattling on and on, I promise.  Just take a look at some of this stuff, and tell me it’s not extremely awesome.  Makes me happy to be alive.

Magic Squares

Add the numbers in any direction, and the add up to the same amount.  Magic squares have been known to people for thousands of years, and they are still amazing today.  I spent a few hours today analyzing the numbers within the squares, and between different kinds of magic squares.  Check this one out. Albrecht Dürer's engraving Melencolia I contains what is believed to be the first magic square seen in European art The 3, 4, 5, 6, 7, and 8 squares

So one of the cool things I’ve discovered is that the average value of each square within a square (e.g. in the 6 square, the average value of each square is 18.5 – 111/6=18.5), compared the the average value of each square in the square one smaller (e.g. the 5 square), is halfway between the number of each of the squares’ rows.  Each row in the 6 square equals 111 – so the average value of each square is 18.5.  In the 5 square, each row adds up to be 65, making the average value per square 13.  18.5-13=5.5.  Great.  When you look at the 5 and 4 squares, the difference in average square value is 4.5.   And 3.5 between the 3 and 4 squares.  Isn’t that amazing?  Why?  What does it mean?

Möbius strip

Fractals Fibonacci Spiral and Golden Ratio  ### 5 Responses

1. on January 13, 2012 at 10:47 pm | Reply אורי

Your discovery can explained as following:

A square of size n contains all the integers in the range 1..n².

The sum of the integers in the range 1..m is given by m(m+1)/2, and dividing this by m to get the average, gives us: (m+1)/2.

Now substitute n² for m, and we get that the average of the numbers in an n-sized square is: (n²+1)/2.

For an (n-1)-sized square, we replace n by n-1, giving us:
((n-1)²+1)/2 = ((n²-2n+1)+1)/2 = (n²-2n+2)/2.

Subtracting this from the value we found for the n-square gives us:
(n²+1)/2 – (n²-2n+2)/2 = (n²+1-(n²-2n+2))/2 = (2n-1)/2.

Which happens to be exactly the average of n and n-1:
(n+(n-1))/2 = (2n-1)/2.

2. on January 15, 2012 at 12:42 pm | Reply Netiva

It’s never too late to become mathematicien, or at least a number theorist:)

Great explanation Uri:)

כל לחי שרון

• on January 26, 2012 at 2:06 pm | Reply Neural Outlet..

“at least” :O You dare speak ill of the Queen of Mathematics! 😛

• on January 27, 2012 at 7:33 pm Irène Sharon Hodes

My mother, to whom you’re replying, is a mathematician… 😉

3. on January 15, 2014 at 3:54 am | Reply Luis E Moscoso

I have analyzed Albrecht Dürer’s Magic Square and playing with numbers myself came with the perfect square that not only adds 34 in any direction, including diagonals and the four corners but also each four connecting squares.
which on Dürer’s square do not add to 34.
My square’s first line is 8 11 14 1
second line 13 2 7 12
third 3 16 9 6
fourth 10 5 4 15
Add for example the four top left corner: 8+11
+ 13 + 2 = 34
Dürer’s top corner is: 16 + 3
+ 9 + 10 = 38
I will appreciate comments on this work.