Questions about the History of an Old Illusion

It all began with a weekend trip to a museum in the small northern Michigan town of Cross Village to do some research on a Franciscan named Father Weikamp who had a convent in the village in the 19th century.  My beautiful Jeannie called me into another room and asked me to look at a picture, shown below, and follow the directions.
The directions were to stare at the dots in the center of the image for 30 seconds and then close your eyes.  In a few moments you see a bright circle with an image in it.  You might try it before we go on, and if my image isn't good enough, you can find lots of links to it on the web.  Here is one.

Ok, So I know this is an after image illusion, and I had seen lots of them, but none that seemed like this one. Here are four from Wikipedia. It seems very easy to me to see what all the others are without focusing on any thing in particular; but the fourth seems not at all like the others.

I was curious and started searching for others that would not be obvious before you stared at the image and closed your eyes. My search was without success. If you know of more references, please advise.
My real question however, was about the scientific history of these types of illusions. The subject seems little studied, and according to several references, we are still unsure if it is in the brain or the eye that the reversal occurs.
The Franciscan mentioned above had died in the 1880's so it has to be a phenomenon that is somewhat old, but tracking down the history was even more difficult.
So now I'm searching for historical information on this particular type of burn in illusion. I would appreciate information resources from anyone who has any knowledge of the history.

The Day America went 100,000,000 Pi

Rats!!! Missed it:



The United States Census has an online “population clock” .
A bit after 2:29 pm EDT, the census bureau said that the United States reached 314,159,265 residents. Notice this is approximately pi * 100,000,000 .



Posted by Tyler Clark on August 14th, 2012 at 5:29 pm at the AMS Graduate Student Blog

Fibonacci Digit Sums

David Brooks sent me a note recently with the following interesting tidbits on the sum of the digits of some Fibonacci numbers.  Thought it was fun, so here they are.  Thanks David:

btw:  here is  OEIS sequence  

Find the next one, and send it here...

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 I ran across this interesting information while researching information on Leon Bankoff.

The sum of the digits of the 5^th Fibonacci number is 5.  (Leon Bankoff) (Ok, the F(5) is five, so that one is a gimme'

The sum of the digits of the 10^th Fibonacci number (55) is 10. (Leon Bankoff)

The sum of the digits of the 31^st Fibonacci number is 31 (1346269).  (Leon Bankoff)

The sum of the digits of the 35^th Fibonacci number is 35.  (Leon Bankoff)

The sum of the digits of the 62^nd Fibonacci number is 62.  (Leon Bankoff)

The sum of the digits of the 72^nd Fibonacci number is 72.  (Leon Bankoff)

The sum of the digits of the 175^th Fibonacci number is 175.  (OEIS)

The sum of the digits of the 180^th Fibonacci number is 180.  (OEIS)

The sum of the digits of the 216^th Fibonacci number is 216.  (OEIS)

The sum of the digits of the 251^st Fibonacci number is 251.  (OEIS)

The sum of the digits of the 252^nd Fibonacci number is 252.  (OEIS)

The sum of the digits of the 360^th Fibonacci number is 360.  (OEIS)

The sum of the digits of the 494^th Fibonacci number is 494.  (OEIS)

The sum of the digits of the 504^th Fibonacci number is 504.  (OEIS)

The sum of the digits of the 540^th Fibonacci number is 540.  (OEIS)

The sum of the digits of the 946^th Fibonacci number is 946.  (OEIS)

The sum of the digits of the 1188^th Fibonacci number is 1188.  (OEIS)

The sum of the digits of the 2222^nd Fibonacci number is 2222.  (OEIS)

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More mature students should read Joshua Zucker's comment below to begin to understand a little more about the frequency with which these occur.

Younger students might think about the digital root of the Fibonacci numbers since that only requires the sum of two simple one digit numbers. 


They might want to explore questions like:
How frequently is the digital root of F(n) the same as the digital root of n (obviously in all the case above, but are there others?)


There be a point where the sequence of digital roots repeats; Why?
 How would that information help you find large numbers for which the digital root of F(n) is the same as the digital root of n? 
Can you see how this would provide a shortcut in the search for numbers in like the sequence David Brooks has written about above?
What is the longest possible string length of one digit numbers before a repeating sequence must emerge?  (Perhaps you might attempt that question with only two digits, zero and one, or three digits and then expand the question).

There are some other interesting things to explore, for instance the sum of the digits of  F(2314) is 2134.