There are two triangles in front of you. The top one is completely divided into four fragments, painted in different colors. In the lower triangle, the same fragments are arranged in a different order. Attention, question. Where did the extra empty square come from? The answer is inside.
In fact, this is not an optical illusion, but an interesting task. The areas of the shaded figures, of course, are equal to each other (32 cells), however, what is visually observed as 13 × 5 triangles, in fact, is not, and has different areas (S13 × 5 = 32.5 cells). That is, the error masked in the condition of the problem is that the initial figure is named a triangle (in fact, it is a concave quad). This is clearly visible in the diagram below - the "hypotenuses" of the upper and lower figures pass through different points: (8.3) at the top and (5.2) at the bottom. The secret is in the properties of the blue and red triangles. This is easy to verify by calculations.
The ratios of the lengths of the corresponding sides of the blue and red triangles are not equal to each other (2/3 and 5/8), so these triangles are not similar, which means they have different angles at the corresponding vertices. Let's call the first figure, which is a concave quadrilateral, and the second figure, which is a concave octagon, pseudotriangles. If the bottom sides of these pseudo-triangles are parallel, then the hypotenuses in both 13×5 pseudo-triangles are actually broken lines (the top picture creates a kink inward, while the bottom picture creates a kink outward). If we superimpose the upper and lower figures 13 × 5 on each other, then a parallelogram is formed between their “hypotenuses”, which contains the “extra” area. In the figure-scheme, this parallelogram is shown in the correct proportions.
The acute angle in this parallelogram is equal to arcctg 46 ≈ 0°1′18.2″. At this angle, the minute hand on a working watch moves in 12.45 s. It is by this amount that the obtuse angle in the parallelogram under consideration differs from the developed one. Visually, such an insignificant difference is imperceptible, but it is clearly visible on the animation.
According to Martin Gardner, this problem was invented by New York amateur illusionist Paul Currie in 1953. However, the principle behind it was known as early as the 1860s. You can see that the lengths of the sides of the figures from this problem (2, 3, 5, 8, 13) are consecutive Fibonacci numbers.
The goat's foot danced
And she drew it.
(circle)
No angle, no side
And relatives are just pancakes.
(a circle )
On our soccer ball
It is painted black.
(pentagon)
Circle the brick with chalk
All on asphalt
And the figure will turn out -
Of course you know her.
(rectangle )
For the figure of this bee
There is no better place in the whole world.
(hexagon)
Two twin squares
Half of their father.
Attach the sides
Say their father's name.
(rectangle )
Angry spatula tail fish
Bit off half a square -
A whole corner, believe it or not!
Who is he, poor thing, now?
(triangle )
Dip in the paint cube,
Apply and lift.
Vasya did this ten times -
They printed.
(squares)
I live well
Above and below.
It's easy for you to guess me -
Five obtuse angles, five acute.
(five pointed star)
The square stood on the corner -
He poked his nose into the ceiling.
Up he grew another five days.
How to call it now?
(diamond)
Dog Mongrel and dog Pirate
Pull the corners of the square.
If they don't stop now
What will they turn into?
(in a rhombus, if pulled at opposite corners)
(in a trapezoid, if for neighboring)
Triangle with a semicircle
Krug was teased as a "fat friend".
Circle, upset to tears,
Already grown up and up.
Who can guess here
What is his name now.
(oval)
Eight flies on the edge of the saucer
Laughing at the spider -
He tied them around
Cobwebs to each other.
There are eight cobwebs too.
Who can name the figure?
(octagon)
Spilled into a notebook Mila
Bright red ink.
"Mila is crying! Mila is a crybaby!” -
Teasing figure - ...
(blot)
Triangle stuck his nose
In a jet vacuum cleaner.
And without a nose, he, - oh, my God! -
Looked like a skirt.
The most interesting thing
What is his name now.
(trapeze)
He is both a ball and a ball,
And the moon, and the bun.
(ball)
Party guest is always,
But not a ball and not a flag,
It looks like the roof of a tower
And a waffle cone.
(cone)
Six squares made friends
And they formed into it forever.
(cube)
“I,” he said early in the morning,
The most wonderful!
I am both a donut and a bagel,
Me and the lifeline!”
(tor)
Here is a brick, a new textbook,
Pastila, a bale of magazines.
Name their shape
Of fourteen letters!
(parallelepiped)
On the floor a square roof
All around the dunes above,
And under her is a pharaoh
His afterlife is watching a dream.
(pyramid)
The mouse lived in the ball, as in a mink,
Ate the pulp all to the crust -
Those are her manners.
The ball turned into...
(sphere)
At the loaf of sausage
We cut off our noses
He is now a man's hat
Amazing beauty.
(cylinder)
Every preschooler wants to know
What is a triangle.
The triangle is a wedge
Triangle trampoline
Rising above the world
Wedge in the sky cranes
fly by
Festive tree -
Also "triangle".
Sail on the sea, all winds
Champion on earth
Looks like a triangle.
And belyash, and the pyramid -
Triangular type.
Sometimes a coffee table
So it looks like a triangle.
Triangular - pomelo,
Triangular - paddle.
Gobbling away
The most appetizing pickle
Dad, mom and child -
There is a family triangle...
(Uglitskikh A.)
triangular triangle
Angular bastard.
It looks like the roof of a house
And on the gnome's hat.
And on the sharp tip of the arrow,
And on the ears of a red squirrel.
Very angular in appearance
It looks like a pyramid!
Triangle - three corners,
Look kids:
Three peaks are very sharp -
Triangle - "sharp-nosed".
It also has three sides:
One, two, three, look.
We draw a triangle
Now we will know him.
The plane flies through the sky
triangle wing,
On my bike
triangular saddle,
There is such an object - a square,
And it's all a TRIANGLE.
Here mom three matches
I put it on the table
And me a triangle
Made from matches.
And at this time I drew
And watched my mom
I connected three lines
And did the same.
I'm under a triangular roof
I'll hide from the rain.
triangular roof,
Hide me quickly!
Rectangle
Like a rectangle window,
Neat, like a schoolboy.
It looks like a door, like books,
And on the boy's backpack.
On the bus, on the notebook,
For a big chocolate.
On the pig's trough
And on a wrapper for a child.
My canvas is not square
It is rectangular!
I will draw on it
Various forms!
We stretched the square
And presented at a glance
Who did he look like
Or something very similar?
Not a brick, not a triangle -
It became a square rectangle.
The mountains are like him.
Similar to a children's slide.
And on the roof of the house
He looks very similar.
What did I guess?
Triangle then, friends.
These figures rush through the city,
Children are taken to kindergartens and schools.
There are buses, trolleybuses, trams.
You look on the road, do not yawn!
Houses look like this figure.
What am I writing about? Answer, friends!
Rhombus
The kite is not easy!
Serpent, it's in the shape of a diamond!
He flies above the ground
Even higher than the house.
The elephant turned the square
Looked at it and sighed.
Sat on top, slightly flattened,
And the square became a rhombus!
Rhombus is a difficult figure,
The two combine:
Triangle one and two
The figure suddenly became alone.
Four sides in a rhombus.
They are equal among themselves.
Four in a rhombus and a corner,
Two are equal to each other.
(Mazhirina N.)
Slightly flattened square
Invites to know:
Acute angle and obtuse
Forever bound by fate.
Guess what's the matter?
How do we name the figure?
Polygon
Six obtuse corners inside
Look at the figure
And imagine that from a square
Got his brother.
Too many corners here
Are you ready to name it?
Trapeze
If you climb higher with a saw,
Saw off the roof of the house
Then we will offend the owners,
But we will see the trapezoid!
And then we'll fix it all
And take out the skirt from the closet.
We'll see: the skirt too
Looks like a trapezoid!
triangle filed
And got the shape:
Two obtuse corners inside
And two sharp ones - look.
Not a square, not a triangle
It looks like a polygon.
The figure is like this
And I don't know her.
Where do you live, trapeze,
America, China?
Maybe behind the trapeze
Should you go to Greece?
Mom says: - Don't,
Trapeze next to you.
I will dispel your sadness
You wait a minute.
And on the ironing board
Putting down her skirt
He runs an iron over it,
So as not to puff up the bag:
- Here's a TRAPEZE for you,
Don't go to Greece.
Cone
Inverted cone -
A vase for flowers.
I'll fill it with water
And the bouquet is ready!
Mom said: - And now
About the cone will be my story.
Stargazer in a high cap
Counts the stars all year round.
CONE - stargazer's hat.
That's what he is. Understood? That's it.
Mom was at the table
She poured oil into bottles.
- Where is the funnel? No funnel.
Look. Don't stand on the sidelines.
- Mom, I'm not going to move,
Tell me more about the cone.
- The funnel is in the form of a watering can cone.
Come on, find me quickly.
I couldn't find the funnel
But mom made a bag,
Wrap cardboard around your finger
And deftly fastened with a paper clip.
Oil is pouring, mom is happy
The cone came out just right.
Cylinder
Cylinder, what is it? I asked my dad.
Father laughed: - Top hat, this is a hat.
To have a correct idea,
The cylinder, let's say, is a tin can.
Steamer pipe - cylinder,
The pipe on our roof - too,
All pipes are similar to a cylinder.
And I gave an example like this -
My beloved kaleidoscope
You can't take your eyes off him.
It also looks like a cylinder.
Figures all remember me
A spyglass will help!
She is a top hat!
And by the way! It has a circle too!
Ball
If the circle is large,
So it's a ball!
Fish! Lunch hour
Arrived in the aquarium!
Hit! Hit! Another hit!
A ball flies into the gate - BALL!
And this is a watermelon ball
Green, round, delicious.
Look better - what a ball!
It is made up of circles.
Cut into circles watermelon
And taste them.
How can we not turn it
There are exactly six equal faces.
We can play loto with him,
Let's just be careful:
He is neither kind nor rude
Because it's a cube.
I'll show you the cube!
What a huge one!
With straight edges!
He is balanced!
The postman brought us a box -
A package for me and my brother.
The box is a CUBE, it has six sides,
All sides are squares.
What is in the package?
There are shavings and sawdust
Candies and bagels,
Also jars with jam.
Pyramid
The Egyptians laid them down
And so cleverly made
What are they worth for centuries.
Guess yourself kids
What are these bodies?
Where is the peak visible to everyone?
Guessed? Because of the look
Everyone knows the pyramid.
I saw the picture. In this picture
There is a PYRAMID in the sandy desert.
Everything in the pyramid is extraordinary,
There is some mystery and mystery in it.
The Spasskaya Tower on Red Square
Both children and adults are well known.
Look at the tower, ordinary in appearance,
What's on top of her? Pyramid!
E. Frantsuzova