Bidimensional display of 31 Rational Numbers by means of the Stern-Brocot Tree [Visualisation bidimensionnelle de 31 Nombre Rationnels à l'aide de l'arbre de Stern-Brocot].

Let's define the so-called Median-Mean of two rational numbers A/B and C/D:

```                                           A+C
MedianMean(A/B,C/D) = -----
B+D
```

Nota: the Median-Mean is at the origin of the so-called Simpson Paradox.

'Left', 'Right' and 'Mean' being three rational numbers, the following recursive algorithm:
```                    Generate(Left,Right)
{
Mean = MedianMean(Left,Right);

Generate(Left,Mean);
Generate(Mean,Right);
}
```

starting with:
```                    zero=0/1
infinity=1/0

Generate(zero,infinity)
```

gives the Stern-Brocot tree where all the positive rational numbers (Q+={Mean}) appear once and only once:
```
level=1   0/1 > > > > > > > > > > > > > > > > > > 1/1 < < < < < < < < < < < < < < < < < < 1/0
|                                     * | *                                     |
|                                   *   |   *                                   |
|                                 *     |     *                                 |
|                               *       |       *                               |
|                             *         |         *                             |
|                           *           |           *                           |
|                         *             |             *                         |
|                       *               |               *                       |
|                     *                 |                 *                     |
level=2   0/1 > > > > > > > > 1/2 < < < < < < < < 1/1 > > > > > > > > 2/1 < < < < < < < < 1/0
|                 * | *                 |                 * | *                 |
|               *   |   *               |               *   |   *               |
|             *     |     *             |             *     |     *             |
|           *       |       *           |           *       |       *           |
level=3   0/1 > > > 1/3 < < < 1/2 > > > 2/3 < < < 1/1 > > > 3/2 < < < 2/1 > > > 3/1 < < < 1/0
|       * | *     * | *     * | *     * | *     * | *     * | *     * | *       |
|     *   |   * *   |   * *   |   * *   |   * *   |   * *   |   * *   |   *     |
(...)     (...)     (...)     (...)     (...)     (...)     (...)     (...)     (...)
```

At last, the colored points display the Rational Numbers with coordinates {X=numerator,Y=denominator} and a luminance proportional to their decimal values. The black dots are on the one hand the Rational Numbers not yet computed or on the other hand the Rational Numbers (NxA)/(NxB) with N>1 that are not computed with the preceding algorithm....

See some related pictures (including this one):

 level=3 level=4 level=5 level=6 level=7 level=8 level=15

 level=3 level=4 level=5 level=6 level=7 level=8

(CMAP28 WWW site: this page was created on 08/05/2022 and last updated on 12/09/2022 12:39:09 -CET-)

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