**Simple Dimension Positiomatics**

Author: Adekola Taylor

November, 2013

Positiomatics was derived from the words "Position and Mathematics". Like regenerative mathematics of distribution of ordered system, Positiomatics is the mathematics of position changing, correlation and location of elements (objects) in an ordered system when subjected to distributive regeneration phenomenon using Logico-Sequential Distribution.

**Order of Positions for Simple Dimension Positiomatics**

**Figure 1 **

From the above the elements (objects) are modally assigned positions by designating them with numbers as presented in the Table 1 below.

Table 1

Designation | Order of Position Ranking | |
---|---|---|

Objects | Vertical Position Ranking (P^{v} ) | Horizontal Position Ranking (P^{h} ) |

7 | 1^{st} | 7^{th} |

4 | 2^{nd} | 4^{th} |

1 | 3^{rd} | 1^{st} |

8 | 5^{th} | 8^{th} |

5 | 5^{th} | 5^{th} |

2 | 6^{th} | 2^{nd} |

9 | 7^{th} | 9^{th} |

6 | 8^{th} | 6^{th} |

3 | 9^{th} | 3^{rd} |

Considering the Order of Positions for Simple Dimension Positiomatics in Table 1 above,
let P^{v}_{dx} = vertical position ranking value at distribution x and P^{h}_{dx} =
horizontal position ranking value at distribution x, therefore for example the horizontal position ranking value
(P^{h}_{dx}) of the object designated by 7 at distribution (dx) = 7^{th}, also its vertical
position ranking value (P^{v}_{dx}) at distribution (dx) = 1^{st} based on position ranking.

So the coordinates of the object designated by number 7 = (7, 1)

For object with designation 4, the coordinates = (4, 2) etc

**Monitoring of Position Changing of each Object (e.g.Card)**

Consider this distribution below (1).

Total number of objects, n(E) = 12 Number of Columns, C = 3 Number of Rows, r = 4

Figure 2

By considering the position changing of objects designated by
numbers 12 and 8, tables of position changing are constructed

**Table 2**:Positional Ranking Values for Object designated by number 12 in the Figure 2

dx |
d0 |
d1 |
d2 |
d3 |
d4 |
d5 |
d6 |
---|---|---|---|---|---|---|---|

P^{v}_{dx} | 9^{th} | 10^{th} | 1^{st} |
4^{th} | 3^{rd} | 12^{th} | 9^{th} |

P^{h}_{dx} | 12^{th} | 9^{th} | 10^{th} |
1^{st} | 4^{th} | 3^{rd} | 12^{th} |

This implies that

At d0, for object 12, the coordinates = (12, 9)

At d1, for object 12, the coordinates = (9, 10)

At d2, for object 12, the coordinates = (10, 1) Etc

**Table 3**:Positional Ranking Values for Object designated by number 8 in the Figure 2

dx |
d0 |
d1 |
d2 |
d3 |
d4 |
d5 |
d6 |
---|---|---|---|---|---|---|---|

P^{v}_{dx} | 6^{th} | 11^{th} | 5^{th} |
7^{th} | 2^{nd} | 8^{th} | 6^{th} |

P^{h}_{dx} | 8^{th} | 6^{th} | 11^{th} |
5^{th} | 7^{th} | 2^{nd} | 8^{th} |

This implies that

At d0, for object 8, the coordinates = (8, 6)

At d1, for object 8, the coordinates = (6, 11)

At d2, for object 8, the coordinates = (11, 5)

At d3, for object 8, the coordinates = (5, 7)

At d4, for object 8, the coordinates = (7, 2)

At d5, for object 8, the coordinates = (2, 8)

At d6 = dt, for object 8, the coordinates = (8, 6)

The graph of the above Table of Positional Ranking Values is given below

**Graph 1**

The slopes of the three lines on the graph are the same

Therefore, slope of the graph line A = dy/dx = -1[4 - 1]/10 - 1 = -1/3 = slope B = slope C

From graph line A

1/4.33 P^{v}_{dx} + 1/13 P^{h}_{dx}= 1

P^{v}_{dx} = -1/3 P^{h}_{dx} + 13/3 .............(1)

From graph line B

1/8.66 P^{v}_{dx} + 1/26 P^{h}_{dx}= 1

P^{v}_{dx} = -1/3 P^{h}_{dx} + 26/3 .............(2)

From graph line C

1/13 P^{v}_{dx} + 1/39 P^{h}_{dx}= 1

P^{v}_{dx} = -1/3 P^{h}_{dx} + 39/3 .............(3)

Equations 1, 2, and 3 are all related and can be rewritten as below.

P^{v}_{dx} = -1/3 P^{h}_{dx} + (13/3) x 1 .............(1)

P^{v}_{dx} = -1/3 P^{h}_{dx} + (13/3) x 2 .............(2)

P^{v}_{dx} = -1/3 P^{h}_{dx} + (13/3) x 3 .............(3)

Therefore, the overall equation for equations 1, 2, and 3 is given below

P^{v}_{dx} = -1/3 P^{h}_{dx} + (13/3)K^{v}_{dx} .............(4)

P^{v}_{dx} = -1/3 P^{h}_{dx} + (4 +1/3)K^{v}_{dx} .............(4)

Where 1 is less than or equal to K^{v}_{dx}, and K^{v}_{dx}is less
than or equal to 3

**Table 4**: Values for K^{v}_{dx}

Graph 1 |
P^{v}_{dx} Class Interval |
K^{v}_{dx}(Vertical position class interval rank) |
---|---|---|

Graph A | 1 - 4 | 1 |

Graph B | 5 - 8 | 2 |

Graph C | 9 - 12 | 3 |

1 is less than or equal to K^{v}_{dx}, and K^{v}_{dx}is less
than or equal to 3

n(E) = 12, C = 3, and r = 4, the overall equation 5 below is gotten by substituting the values of 3 and 4 with C and
r respectively into equation 4

P^{v}_{dx} = -1/C P^{h}_{dx} + (r + 1/C)K^{v}_{dx} .............(5)

Where 1 is less than or equal to K^{v}_{dx}, and K^{v}_{dx}is less than or equal to C

Any distribution in simple dimension positiomatics obeys the above formulas for the relationship
between P^{v}_{dx} and P^{h}_{dx} in distributive regeneration of ordered system.

**Illustration**

In a ceremony, 12 pupils are needed to perform choreography to grace the occasion. They are grouped into 3 columns with
each column having 4 pupils. Moreover their position changing obeys Logico-sequential distribution. If a particular
pupil has his vertical position ranking value P^{v}_{d4} = 4^{th}, calculate his horizontal
position ranking value P^{h}_{d4} at that particular distribution.

**Solution**

P^{v}_{dx} = -1/C P^{h}_{dx} + (r + 1/C)K^{v}_{dx}

Total number of pupils, n(E) = 12, number of column, {C} = 3 number of rows, (r) = 4

P^{v}_{d4} = 4^{th}

P^{v}_{d4} = -1/3 P^{h}_{d4} + (4 + 1/3)K^{v}_{d4}

Where 1 is less than or equal to K^{v}_{d4}, and K^{v}_{d4}is less than or equal to C

Calculation of K^{v}_{d4} (Vertical position class interval rank)

n(E)/C = 12/3 = 4

Pupils in class interval (1 - 4) have their K^{v}_{dx} = 1, pupils in class interval (5 - 8)
have their K^{v}_{dx} = 2, and pupils in class interval (9 - 12) have their
K^{v}_{dx} = 3

Since that particular pupil has his vertical position ranking value P^{v}_{d4} = 4, then he falls into
class interval (1 - 4) having his K^{v}_{d4} = 1

P^{v}_{dx} = -1/3 P^{h}_{d4} + (4 + 1/3)1

It gives, P^{h}_{d4} = 3

The horizontal position ranking value P^{h}_{d4} of that particular pupil at that d4
(distribution 4) = 3.

In other words, at d4 that particular pupil would be at 3rd position horizontally.

**Problems**

- 50 cards are grouped into 10 columns and 5 rows with each column having 5 cards. If a card at d4 (distribution 4)
has its vertical position, P
^{v}_{d4}= 2^{nd}. Find the horizontal position P^{h}_{d4}of the card. - A student wanted to perform a card mathematical magic with 1000 cards
which were to be grouped into 10 columns with each column having 100 cards. If his plan was to put the spectator's card
at the vertical position ranking value P
^{v}_{d0}= 125 at the starting distribution (d0). What would be the corresponding horizontal position value (P^{h}_{d0}) of the card at starting distribution (d0). - An ordered system with n(E) = 72 cards can be expressed as a product of C[C+1], where C is the number of
columns and (C+1) is the number of rows. Each column contains equal number of cards. If a card has its
P
^{v}_{d3}= 31^{st}. Find P^{h}_{d0}.

**There are also formulas for the following:**

- Formula for the relationship between P
^{h}_{dx}and P^{v}_{dx} -
Formula for the relationship between P
^{h}_{dx}and P^{h}_{d(x+1)} - Formula
for the relationship between P
^{v}_{dx}and P^{v}_{d(x+1)} - Formula for the
relationship between P
^{h}_{dx}and P^{h}_{d(x+1)}, if n(E) = C^{x} - Formula for the relationship between P
^{h}_{dx}and P^{h}_{d(x+2)}, if n(E) = C^{x} - Formula for the relationship between P
^{v}_{dx}and P^{v}_{d(x+2)}, if n(E) = C^{x} - Formula for the relationship
between P
^{h}_{dx}and P^{h}_{d(x+2)}, if n(E) = C[C + 1] - Formula for the
relationship between P
^{v}_{dx}and P^{v}_{d(x+2)}, if n(E) = C[C + 1] etc - Taylor, Adekola A. (2016). Derivation of formulas for position change of entities in power inductive distribution.
*International Journal of Scientific and Research Publications*6(7):409-420. - Taylor, Adekola A. (2015). Quadratic distribution patterns in Kola analysis.
*Mathematical Theory and Modeling*5: 60-66. - Taylor, Adekola A. (2015). Derivation of formulas for position change in Kola analysis.
*International Journal of Scientific and Research Publications*5(11):101-109. - Taylor, Adekola A. (2014).
*Logical-mathematical intelligence for teens.*Mathsthoughtbook.com - Taylor, Adekola A. (2013).
*Regenerative mathematics and dimurelo puzzles for children 8-12yrs.**USA: Lulu Press Inc.* - Taylor, Adekola (2013).
*Card magic and my mathematical discoveries.*USA:Lulu Publishing. - Taylor, Adekola A. (2010). Kola Analysis: An inventive approach to logical-mathematical intelligence for secondary and advanced levels.
*Journal of Mathematical Sciences Education*1(1): 44-53.

**See Also:****Regenerative Mathematics;****Card Mathematical Intelligence Games;****Dimurelo Puzzles;****Logical-Reasoning Puzzles for Children;****Mathematical Discoveries;****Abstract-: Kola Analysis: An Inventive Approach to Logical-Mathematical Intelligence for Secondary and Advanced Levels****Dreams, Visions, and Empowerment;****Multiple Intelligences;****The Etiology of IQ;****The Bane of Plagiarism;****Stop Teen Cosmetic Surgery;****Human Development and Aging;****Thoughts;****How to Stop All Your Bad Habits in just 21 Days;****Thinking for You Series;****The Black People: We Need A Change;****Logical-Mathematical Reasoning for Teens;****Regenerative Mathematics and Dimurelo Puzzles for Children;****The Thoughts You Need to Think;****The Foundation of American Constitution: A Lesson for All**

**References**

- Formula for the relationship between P

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