25 April 2018

Casa da Música


Casa da Música in Porto, Portugal, a concert hall designed by the Dutch architect Rem Koolhaas, was opened on \(15\) April \(2005\).

24 April 2018

New York City I by Piet Mondrian


New York City I (\(1942\)) is a painting by the Dutch painter Piet Mondrian (\(1872 - 1944\)).

23 April 2018

Hypotrochoid

The hypotrochoid is a transcendental plane curve, corresponding to a fixed point on a moving circle that rolls without slipping over and internally around another circle, called the director circle. This curve can be drawn through the spirograph.

The parametric equations for a epitrochoid are:

\(x=\left( R-r \right) \cos t+d \cos \left( \frac{R-r}{r}t \right)\)

\(y=\left( R-r \right) \sin t-d \sin \left( \frac{R-r}{r}t \right)\)

where \(R\) is the radius of the director circle (fixed), \(r\) is the radius of the mobile circle, \(d\) is the distance from the center point of the mobile circle and \(t\) is the parameter of the angle.

The hypocycloid (\(d=r\)) and the ellipse (\(R=2r\)) are particular cases of the hypotrochoid.

In the applet below, click in the Start button to begin the animation that shows the drawing of an hypotrochoid. You can use the Stop button to stop the animation at any point. Click in the Reset  button to erase the last curve. You can change the values of R, r and d in the sliders in order to obtain different kinds of hypotrochoids. You can also hide the circles and the line segment in order to see only the curve.

22 April 2018

One half

Some mathematical operations whose result is \(\frac{1}{2}\):

  • \(\cos 60º = \frac{1}{2}\);
  • \(\frac{\cos 30º}{\sqrt 3}=\frac{1}{2}\);
  • \(\frac{1}{3}+\frac{1}{6}=\frac{1}{2}\).
Using all the digits from \(1\) to \(9\):
  • \(\frac{6,729}{13,458}=\frac{1}{2}\).
Using the same number twice but changing the position of one digit:
  • \(\frac{105,263,157,894,736,842}{210,526,315,789,473,684}=\frac{1}{2}\).
Numbers with such property are called parasitic numbers.

21 April 2018

Möbius

The Möbius strip is a surface that has only one side and only one boundary component. It has the mathematical property of being non-orientable. The Moebius strip was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in \(1858\).

A model can easily be created: cutting a strip, making it a half-twist and glue up the ends, giving its characteristic shape.

The Möbius strip has several curious properties. Cutting the strip in half longitudinally of this will appear another strip (it is not a Möbius strip because it has two sides) but with twice the length of the original. Cutting the strip longitudinally about a third of their width, we obtain two strips engaged, one smaller than the other. The smallest is a Möbius strip and the biggest is not.

The Möbius strip has many applications, especially in architecture, design and engineering.

\(\left(-0.99\left(x\left(x^2+y^2-z^2+1\right)-2yz\right)-2.03\left(x^2+y^2\right)\right)^2=\left(x^2+y^2\right)\left(1.01\left(x^2+y^2+z^2+1\right)-1.98\left(yz-x\right)\right)^2\)

20 April 2018

1089

Choose a three-digit number so that the first and third digits differ two or more units:
  1. Write the number from right to left;
  2. Do the subtraction of the larger number by the smaller;
  3. Write the number obtained in point 2 from right to left and add it to the number also obtained in point 2.
And the result is \(1089\).

Example: Consider the number \(503\).
  1. \(305\)
  2. \(503-305=198\)
  3. \(198+981=1089\)

19 April 2018

Sudoku


Sudoku is a game in a grid (square of dimensions \(9 \times 9\)) designed in \(1979\) by the American Howard Gams, inspired by the Latin square, as well as the problem of the \(36\) officers of the Swiss mathematician Leonhard Euler.
The game objective is to complete the grid with a series of digits, all different, which can not be repeated nor in the same row, nor in the same column and nor in a sub-grid (square of dimensions \(3 \times 3\)). Some numbers are already in the grid, which allows the progressive resolution of the game. You can also play Sudoku online.