Bidimensional display of 255 Rational Numbers by means of the Stern-Brocot Tree [Visualisation bidimensionnelle de 255 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 {1/1,1/2,1/3,1/4,1/5,1/6,1/7,1/8,2/13,2/11,3/17,3/16,2/9,3/14,4/19,5/23,3/13,5/22,4/17,2/7,3/11,4/15,5/19,7/26,5/18,8/29,7/25,3/10,5/17,7/24,8/27,4/13,7/23,5/16,2/5,3/8,4/11,5/14,6/17,9/25,7/19,11/30,10/27,5/13,8/21,11/29,13/34,7/18,12/31,9/23,3/7,5/12,7/17,9/22,12/29,8/19,13/31,11/26,4/9,7/16,10/23,11/25,5/11,9/20,6/13,2/3,3/5,4/7,5/9,6/11,7/13,11/20,9/16,14/25,13/23,7/12,11/19,15/26,18/31,10/17,17/29,13/22,5/8,8/13,11/18,14/23,19/31,13/21,21/34,18/29,7/11,12/19,17/27,19/30,9/14,16/25,11/17,3/4,5/7,7/10,9/13,11/16,16/23,12/17,19/27,17/24,8/11,13/18,18/25,21/29,11/15,19/26,14/19,4/5,7/9,10/13,13/17,17/22,11/14,18/23,15/19,5/6,9/11,13/16,14/17,6/7,11/13,7/8,2/1,3/2,4/3,5/4,6/5,7/6,8/7,13/11,11/9,17/14,16/13,9/7,14/11,19/15,23/18,13/10,22/17,17/13,7/5,11/8,15/11,19/14,26/19,18/13,29/21,25/18,10/7,17/12,24/17,27/19,13/9,23/16,16/11,5/3,8/5,11/7,14/9,17/11,25/16,19/12,30/19,27/17,13/8,21/13,29/18,34/21,18/11,31/19,23/14,7/4,12/7,17/10,22/13,29/17,19/11,31/18,26/15,9/5,16/9,23/13,25/14,11/6,20/11,13/7,3/1,5/2,7/3,9/4,11/5,13/6,20/9,16/7,25/11,23/10,12/5,19/8,26/11,31/13,17/7,29/12,22/9,8/3,13/5,18/7,23/9,31/12,21/8,34/13,29/11,11/4,19/7,27/10,30/11,14/5,25/9,17/6,4/1,7/2,10/3,13/4,16/5,23/7,17/5,27/8,24/7,11/3,18/5,25/7,29/8,15/4,26/7,19/5,5/1,9/2,13/3,17/4,22/5,14/3,23/5,19/4,6/1,11/2,16/3,17/3,7/1,13/2,8/1} with coordinates {X=numerator,Y=denominator} and a luminance proportional to their decimal values.



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level=3

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level=3

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level=8




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