This page was originally posted for discussion within Oregonians for Rationality of that organization's booth at da Vinci Days celebration in Corvallis, Oregon, July 15-16, 2000.   It is now being linked from our Web site because of its general interest.
Some of the things we plan to bring to Da Vinci Days Corvallis, OR; 2000

This object can easily be made by cutting a card and then folding it.  If you don't image three dimensional relationships fairly well, it can look impossible.  We plan to have pieces of stiff paper and scissors so people can try to do it.


These six blocks can be packed into the box with space left over and with nothing sticking out of the box.  It's surprisingly tricky.


How many blocks?  
What are their dimensions?

These blocks can be packed into a larger cubical box with no spaces unoccupied and nothing sticking out.  This looks very easy.  It is not!  Both this puzzle and the previous can be reasoned; it's not just a matter of trial and error.
Experiences at Da Vinci Days
The cutting and folding of the card proved a challenge to all ages, and adults were not particularly better at it than the younger visitors.

The simple mathematical principle which easily leads to a solution to the two box-packing puzzles was seen as obvious by very few of the visitors, but some discovered it with a little guidance.  The smaller puzzle was easily solved by many; the larger may not have been solved by anyone without help.  (One person worked on it for many hours.)  This is a remarkable puzzle which demonstrates the value of reasoning.  We might provide a brief description of how to make it. [DONE!  Click on the picture of it above.]  (It came from an article in Scientific American.) 

We distributed a few printed descriptions of the Steinhaus version of the Soma cube puzzle:

Click here to discover the secret...
This is thought to be the most puzzling version.  However, we have found a surprising and tricky way to make it even more puzzling and interesting.  Our written description explains how it works.  A great construction project for the puzzle enthusiast, especially the devious one.

The following is an image of a familiar object.  However, because we recognize shapes by their edges and the edges have been removed from this image, we don't see the object.  The edge is the boundary between the smooth dots and the irregular dots.  By drawing that boundary in, we can see the object.  We will laminate some dot drawings in plastic so that people can draw on the plastic in erasable felt pen.


Experiences at Da Vinci Days
The smooth dot-irregular dot patterns were enjoyed by even the younger visitors.  Here is something the youngest can do and then see.

Note the photo of the dalmation below.  It's where these patterns had their beginning.

Posters illustrating visual phenomena:

June 22, 2000
The two "Human Vision" posters on the left illustrate the importance of edges to our vision.  Up close, the portrait at the top is just a bunch of tally marks. 
(Lower poster) The eye is almost unrecognizable up close: it's a half-tone of huge dots.  The gargoyle (on the left) is composed entirely of little icons: close up, the gargoyle is completely unrecognizable.  The dalmatian is the famous photo that illustrates what happens when edges are taken away.

The three posters on the right:
The two eyes are from our Web site, "The three Cyclops of ancient mythology...They are watching you.  Do you see them?"

"See like a giant!" illustrates how climbers can find routes using large-base stereoscopic pairs of photos.  Arches National Park's Old Maid's Bloomers (sometimes called "Delicate Arch") illustrates the distinction between eye-crossing and eye-spreading for unaided stereoscopic viewing.

The crystal model photos illustrate how we make something look very big by taking a stereoscopic pair with a very small inter-camera distance.

This is a large sheet of tiles made into a piece of wallpaper.  The tiles are random-dot stereograms, six of which are show on the right.  When we approach a repeating pattern such as this, we frequently fuse adjacent tiles—or further.  This is "the wallpaper effect." When different tiles are fused, we see a stereoscopic pattern within the tiles.  Most people who perceive the patterns are surprised and reach out to touch the hovering objects and the holes.  When their hand enters their field of view, correct fusion occurs and the 3-D objects suddenly vanish.  This usually startles.

Stereo photography is well known to require two separate views of the same object, taken a slight distance apart.  That isn't quite true.  A single photograph can result in a truly stereoscopic image—if the object repeats (like a crystal structure) left to right.  The photo must be viewed by eye crossing.  In this image the eyes should be crossed to fuse two wires away.
Experiences at Da Vinci Days
Most visitors would have benefited from looking at some of the stereoscopic images through a stereoscope, especially if no one was present to explain these images.  Many were able to see them with the unaided eye (those stereoscopic posters have introduced the technique to a lot of people), but most of those who did needed to be guided through the process.

We are gathering the pieces to do Martin Gardner's checkerboard and dominoes puzzle.  See it.

Experiences at Da Vinci Days
The checkerboard and dominoes puzzle was a challenge to all ages.  Again, many of the younger visitors got it and many of the adults didn't.  It's sister puzzle, Martin Gardner's buzz saw cutting of the cube into 27 smaller cubes got very little attention:  it was tucked in the back of the booth on a wordy poster.  It should be presented along with a pile of 27 smaller cubes. See it.

These two puzzles are great exemplars for the concept of logical imperative and a model for the idea that some simple things simply cannot be denied.  When they are denied, the one doing the denying simply is not "seeing" some subtle subtext of the question.

Here are three very special items.  Ask yourself, "What insights could have great influence on causing better problem solving if only more people could see the world through them?"  Here are three we think would have as great an influence for a better functioning society as any.  They are very simple.  They are very subtle.  They are very valuable.

Posters illustrating valuable ideas:
(food for the imagination)

Poster #1



Who cares?
If it’s the basketball coach, he means who’s the tallest.
If it’s the football coach, me means who’s the heaviest.
If it’s the rescuer needing help to lift the beam up, he means who’s the strongest.
If it’s the Mafia boss wanting a bodyguard, he means who’s the most skilled fighter.
Then, who’s the smartest?
Who cares?
If it’s the spelling bee judge, she mean who has the biggest memory and best recall.
If it’s the corporate executive wanting a secretary, she means who’s the best organized.
If it’s the army general wanting someone to fight the battle, he means who’s the most skilled with computers.
If it’s the head of state needing to solve a political problem, he means who can deal with the most parameters.
If it’s the chief engineer, she means . . . ???
Whoever, or whatever,
setting people up along a line doesn’t compare them usefully.

They must be considered in a space of many dimensions. 


….always…. depends...
Line 'em up


Comparatives and superlatives almost always mean something different

The only way to line those ellipse people up by size and make it look right, is to put them into two dimensions. 

This is because ellipses have both height and width (major axis and minor axis). 

That’s two parameters. 

So a line doesn’t work.  It takes a space—two dimensions.

from what you imagined.

This is the issue of rank ordering, unique rank order.  Now, that's something that is almost always "seen."  But It's actually something that is almost never there.  It's the same point we make with ordering colors (instead of ellipses).  It has significance in just about every aspect of daily living:  Who's "worth" more than who?  Who does better than who in school?  Where is the best place to live?  Which is the best car to buy?  Who's the best doctor?

The truth is always, "It depends, it depends..."

(The following html "poster" has a layout quite different from the actual poster we've printed. The printed poster has no background colors and many more graphics.)

Poster #2

‘s wonderful!
‘s marvelous!!
It's true!!!

You, too, can prove* anything your heart desires.  It’s easy.  Just…

Glean confirmations
Ignore disconfirmations

That's the Prove Anything Ploy(PAP)
A flight of raptors are flying above.
One of them comments: 
"Hey guys!  There’s a gull who ignores
his disconfirmations.  Let’s take him!"


He was elated as he examined his lottery tickets in the supermarket checkout line.  The thrill of winning shone in his face.  The desire to win radiated from him through animated body language.  The iron will to ignore his losses roared out, too, in a language of gullibility unheard by himself, but screechingly loud and clear to the lady standing in line behind him. 

"Good grief!" she thought to herself.  "Another supermarket supersucker!" 

"I'll reinvest it," said the man to the checkout clerk.  "Give me two more tickets.

What are
the odds?

If the odds are:

What do
they mean?
1 in 100
You would be wise to bet.  Winning is a sure thing.  (You probably rigged the game.) 
1 in 101
Not a good bet.  Unless you get, at the very least, ten times what you anteed into the pot.
1 in 102
Terrible odds!  Better find another game. 
1 in 103
You’re getting downright gullible if you play this game. 
1 in 104
What, me worry!
1 in 105
A fool’s paradise
1 in 106
You’re more likely to die in an auto accident during the year.

The odds of winning with a lottery ticket is about
1 in 107 to 108
Somebody wins each of those lotteries.  It could be me. 

How many times the cost of my ticket do my winnings have to be before the odds become favorable to me?  Otherwise how certain is my losing?

The first tool of the consummate liar?

Look again!

Statistics is the modern human tool for predicting the future.  It’s the best we’ve got for facing the realities of chaotic events, multiple influences, and uncertainties of life.


Statistics is the first tool of the consummate swift out to prey upon the ultimate gull.  We can avoiding becoming a gull by taking the second glances that the swift has discovered. Be a swift, not a gull!

Statistics is the first tool of the consummate scientist facing the uncertainties of the world.  Math is a marvelous tool box of deep-looking second glances.

So look again  ...and again  ...and again ...and again  ...and again  ...and again… ...and again ..........

This is the issue of "The truth, the whole truth, and nothing but the truth."  The phrase is familiar; the concept underlying it is not.  "The Seven (plus one) Tools of Propaganda" are effective, in part, because this abstract concept is seldom seen very clearly.  The fundamental statistical nature of things around us is usually even less clearly seen.
(Look for the "LOOK OUT!!..." hidden link)


(As above, the following html poster differs from the printed poster:)
Poster #3

The Sun's Secret
What causes sunburn?

When can it happen?

Why does sunbathing sometimes never produce a suntan?
What does a bee know that you don’t?
How can you “see” how actinic the sunlight is?

[From each of the following captions
(in boxes), arrows point to the appropriate
spectral range.]
You see this range
Skin damage comes from here (the actinic range)
A bee sees this range

However, if you are protanopic (a colorblindness), you see this range.
And you can arrange all the colors you see in a two dimensional array.


A bee sees what we can’t see but need to see for avoiding sunburn.  It took our ancestors many millennia to figure out how to “see” beyond our human perception.

The secret of the sun is there for all to see.  Mere humans must peer past the edges of (easy) human comprehension to see her secrets.

Arizona newspapers print daily data that you need to discover one of the sun’s secrets.  But you have to add something almost nobody thinks to add: the angle of the sun (here given from the zenith).  Then you have to  sort through the mess of parameters and pick the relevant from the irrelevant.
(solar noon = 12:09)
9am    69.0
10am  54.7
11am  54.7
12       52.1
1pm    53.5
2pm    58.4
3pm    66.2
4pm    75.9

Sun angles from the zenith, for Tucson, AZ, where the instrument used is located.

(solar noon = 12:29)
9am   47.1
10am  34.5
11am  22.2
12       11.9
1pm    12.0
2pm    22.5
3pm    34.8
4pm    47.5

Table printed in many Arizona newspapers.

When the sun gets much below about 45º from the horizon, almost all of the actinic ultraviolet is gone.  You can use the length of your shadow on a horizontal surface to get a quick and easy measure of the sunlight’s actinicity. If your shadow is longer than you are tall, there’s very little ultraviolet.

The same atmospheric effect that makes the sky blue scatters the uv into outer space: not very effectively when the sun is high; but when the sun is low...there goes the uv.  (It’s called Rayleigh scattering)

Avoiding sunburn on this Web site.

Identifying and sorting through multiple influences is one of the big differences between science and non-science, between "rationality" and nonsense.  The insights needed are what Jean Piaget identified as the most sophisticated and advanced mental developments that occur during normal human development (formal operations).  When they are seen, they are obvious; when not seen, they seem inconceivable.

Go to "Look Again"
(our da Vinci Days handout)