# Did you know that...

π=3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 ...

- A sphere has two sides. However, there are one-sided surfaces.
- There are shapes of constant width other than the circle. One can even drill square holes.
- There are just five regular polyhedra
- In a group of 23 people, at least two have the same birthday with the probability greater than 1/2
- Everything you can do with a ruler and a compass you can do with the compass alone
- Among all shapes with the same perimeter a circle has the largest area.
- There are curves that fill a plane without holes
- Much as with people, there are irrational, perfect, complex numbers
- As in philosophy, there are transcendental numbers
- As in the art, there are imaginary and surreal numbers
- A straight line has dimension 1, a plane - 2. Fractals have mostly fractional dimension
- You are wrong if you think Mathematics is not fun
- Mathematics studies neighborhoods, groups and free groups, rings, ideals, holes, poles and removable poles, trees, growth ...
- Mathematics also studies models, shapes, curves, cardinals, similarity, consistency, completeness, space ...
- Among objects of mathematical study are heredity, continuity, jumps, infinity, infinitesimals, paradoxes...
- Last but not the least, Mathematics studies stability, projections and values, values are often absolute but may also be extreme, local or global.
- Trigonometry aside, Mathematics comprises fields like Game Theory, Braids Theory, Knot Theory and more
- One is
*morally*obligated not to do anything impossible - Some numbers are square, yet others are triangular
- The next sentence is true but you must not believe it
- The previous sentence was false
- 12+3-4+5+67+8+9=100 and there exists at least one other representation of 100 with 9 digits in the right order and math operations in between
- One can cut a pie into 8 pieces with three movements
- Program=Algorithms+Data Structures
- There is something the dead eat but if the living eat it, they die.
- A clock never showing right time might be preferable to the one showing right time twice a day
- Among all shapes with the same area circle has the shortest perimeter
- Curves of infinite length may enclose finite areas.
- Falsity implies anything.
- There is order in chaos.
- To get
*cafe au lait*one should carry coffee to milk and not milk to coffee. - Sets may be thick, thin and normal.
- In some circumstances index equals the content.
- In other circumstances, an index may have a content of its own.
- There are things distant yet near. There are others that are near yet distant.
- There are three plane regions that share exactly the same boundary.
- A continuous additive function must have the form f(x)=ax. Discontinuous linear functions look dreadful.
- A continuous function may grow considerably virtually without changing.
- You can't add apples and oranges but you can add their shapes.
- There are many things that can be added: numbers, vectors, matrices, spaces, shapes, sets, functions, equations, strings, chains...
- Among any two integers or real numbers one is larger, another smaller. But you can't compare two complex numbers.
- The only triangle with rational sides and angles is equilateral.
- 0!=1
- One is
*morally*obligated to do everything impossible. - The word 'fraction' derives from the Latin fractio - to break. However, there are continuous fractions.
- For every object there is a distance at which it looks its best.
- At any given time in New York there live at least two people with the same number of hairs.
- Sometimes in order to add one has to take the difference.
- Demographic tests show that the person least likely to buy
Wired magazine is an American
schoolteacher.
- Complex numbers are in a sense perfect while there is little doubt that perfect numbers are complex.
- You can position 10 defenders of a square castle so that on every side there will be 5 men.
- There are many things that can be multiplied: numbers, vectors, matrices, functions, equations, sets, pegs...
- A torus may be brushed smooth but a sphere can not.
- A circle may be quite useful in drawing straight lines.
- In the sequence of all integers, there are arbitrary long runs with no primes.
- With just one caveat, anything you can do with a compass and a ruler you can do with the ruler alone.
- There are really impossible things.
- You can add apples and oranges.
- Complex number to a complex power may be real.
- Irrational number to an irrational power may be rational.
- There are trisectable angles that are not constructible.
- There exist triangular numbers that are also square.
- No two integers are equidistant from the square root of 2
- Almost every integer has a digit 3 in it
- C
_{0}- C_{0}= [-1, 1] - The length of the diagonal of the unit square equals the square root of 2
- Every composite number is the product of some factors and also the some of the same numbers
- Simple quadrilaterals tessellate the plane
- There is a simple solution to the affirmative action problem
- Two simple polygons of equal area can be dissected into a finite number of congruent polygons
- cos(36°) = (1 + √5)/4
- 1/3 + 1/4 = 7/12
- Σ2
^{-n}= Σn·2^{-n} - Bisector of an imaginary angle may be real
- Any infinite set contains uncountably many nested subsets
- Some numbers are lucky. 13 is one
- How to write an equation of the union of two sets
- Altitudes have ears, foot, stem, and root

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