Simple Dimension Positiomatics

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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 (Pv )Horizontal Position Ranking (Ph )
7 1st7th
4 2nd4th
1 3rd1st
8 5th8th
5 5th5th
2 6th2nd
9 7th9th
6 8th6th
3 9th3rd

Considering the Order of Positions for Simple Dimension Positiomatics in Table 1 above, let Pvdx = vertical position ranking value at distribution x and Phdx = horizontal position ranking value at distribution x, therefore for example the horizontal position ranking value (Phdx) of the object designated by 7 at distribution (dx) = 7th, also its vertical position ranking value (Pvdx) at distribution (dx) = 1st 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
Pvdx9th10th1st 4th3rd12th9th
Phdx12th9th10th 1st4th3rd12th

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
Pvdx6th11th5th 7th2nd8th6th
Phdx8th6th11th 5th7th2nd8th

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 Pvdx + 1/13 Phdx= 1
Pvdx = -1/3 Phdx + 13/3 .............(1)
From graph line B
1/8.66 Pvdx + 1/26 Phdx= 1
Pvdx = -1/3 Phdx + 26/3 .............(2)
From graph line C
1/13 Pvdx + 1/39 Phdx= 1
Pvdx = -1/3 Phdx + 39/3 .............(3)
Equations 1, 2, and 3 are all related and can be rewritten as below.
Pvdx = -1/3 Phdx + (13/3) x 1 .............(1)
Pvdx = -1/3 Phdx + (13/3) x 2 .............(2)
Pvdx = -1/3 Phdx + (13/3) x 3 .............(3)
Therefore, the overall equation for equations 1, 2, and 3 is given below
Pvdx = -1/3 Phdx + (13/3)Kvdx .............(4)
Pvdx = -1/3 Phdx + (4 +1/3)Kvdx .............(4)
Where 1 is less than or equal to Kvdx, and Kvdxis less than or equal to 3
Table 4: Values for Kvdx

Graph 1 Pvdx Class Interval Kvdx(Vertical position class interval rank)
Graph A1 - 4 1
Graph B5 - 8 2
Graph C9 - 12 3

1 is less than or equal to Kvdx, and Kvdxis 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
Pvdx = -1/C Phdx + (r + 1/C)Kvdx .............(5)
Where 1 is less than or equal to Kvdx, and Kvdxis less than or equal to C
Any distribution in simple dimension positiomatics obeys the above formulas for the relationship between Pvdx and Phdx 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 Pvd4 = 4th, calculate his horizontal position ranking value Phd4 at that particular distribution.
Solution
Pvdx = -1/C Phdx + (r + 1/C)Kvdx
Total number of pupils, n(E) = 12, number of column, {C} = 3 number of rows, (r) = 4
Pvd4 = 4th

Pvd4 = -1/3 Phd4 + (4 + 1/3)Kvd4

Where 1 is less than or equal to Kvd4, and Kvd4is less than or equal to C
Calculation of Kvd4 (Vertical position class interval rank)
n(E)/C = 12/3 = 4

Pupils in class interval (1 - 4) have their Kvdx = 1, pupils in class interval (5 - 8) have their Kvdx = 2, and pupils in class interval (9 - 12) have their Kvdx = 3

Since that particular pupil has his vertical position ranking value Pvd4 = 4, then he falls into class interval (1 - 4) having his Kvd4 = 1

Pvdx = -1/3 Phd4 + (4 + 1/3)1
It gives, Phd4 = 3

The horizontal position ranking value Phd4 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

  1. 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, Pvd4 = 2nd. Find the horizontal position Phd4 of the card.

  2. 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 Pvd0 = 125 at the starting distribution (d0). What would be the corresponding horizontal position value (Phd0) of the card at starting distribution (d0).
  3. 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 Pvd3 = 31st. Find Phd0.

    There are also formulas for the following:
    1. Formula for the relationship between Phdx and Pvdx

    2. Formula for the relationship between Phdx and Phd(x+1)

    3. Formula for the relationship between Pvdx and Pvd(x+1)

    4. Formula for the relationship between Phdx and Phd(x+1), if n(E) = Cx

    5. Formula for the relationship between Phdx and Phd(x+2), if n(E) = Cx

    6. Formula for the relationship between Pvdx and Pvd(x+2), if n(E) = Cx

    7. Formula for the relationship between Phdx and Phd(x+2), if n(E) = C[C + 1]

    8. Formula for the relationship between Pvdx and Pvd(x+2), if n(E) = C[C + 1] etc

    9. References

      1. 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.

      2. Taylor, Adekola A. (2015). Quadratic distribution patterns in Kola analysis. Mathematical Theory and Modeling 5: 60-66.

      3. Taylor, Adekola A. (2015). Derivation of formulas for position change in Kola analysis. International Journal of Scientific and Research Publications 5(11):101-109.

      4. Taylor, Adekola A. (2014). Logical-mathematical intelligence for teens. Mathsthoughtbook.com

      5. Taylor, Adekola A. (2013). Regenerative mathematics and dimurelo puzzles for children 8-12yrs. USA: Lulu Press Inc.

      6. Taylor, Adekola (2013). Card magic and my mathematical discoveries. USA:Lulu Publishing.

      7. 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 ELECTRONIC ARTS, INC. (Origin Store)

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