Bidimensional display of 127 Rational Numbers by means of the Stern-Brocot Tree [Visualisation bidimensionnelle de 127 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 {1/1,1/2,1/3,1/4,1/5,1/6,1/7,2/11,2/9,3/14,3/13,2/7,3/11,4/15,5/18,3/10,5/17,4/13,2/5,3/8,4/11,5/14,7/19,5/13,8/21,7/18,3/7,5/12,7/17,8/19,4/9,7/16,5/11,2/3,3/5,4/7,5/9,6/11,9/16,7/12,11/19,10/17,5/8,8/13,11/18,13/21,7/11,12/19,9/14,3/4,5/7,7/10,9/13,12/17,8/11,13/18,11/15,4/5,7/9,10/13,11/14,5/6,9/11,6/7,2/1,3/2,4/3,5/4,6/5,7/6,11/9,9/7,14/11,13/10,7/5,11/8,15/11,18/13,10/7,17/12,13/9,5/3,8/5,11/7,14/9,19/12,13/8,21/13,18/11,7/4,12/7,17/10,19/11,9/5,16/9,11/6,3/1,5/2,7/3,9/4,11/5,16/7,12/5,19/8,17/7,8/3,13/5,18/7,21/8,11/4,19/7,14/5,4/1,7/2,10/3,13/4,17/5,11/3,18/5,15/4,5/1,9/2,13/3,14/3,6/1,11/2,7/1} with coordinates {X=numerator,Y=denominator} and a luminance proportional to their decimal values.



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 08/14/2022 and last updated on 12/09/2022 12:39:19 -CET-)



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