Regular and Repeating Groups

Groups are constituted by consecutive variations.

The size of the group is the number of variations in it
(except for flats - groups of equal gaps).

  All regular groups have symmetry, axial or polar.

Symmetrical groups having any variations:
symmetriads having a horisontal axis of symmetry,

curls                having polar symmetry;


Special groups

stairs   having equal variations
a special kind of curls);
zigzags     having the same absolute values and alternating   signs of consecutive variations
(a special kind of even symmetriades or odd curls)
                and flats having only variations equal zero,
                               and which size of is the number of consecutive constant gaps in it
(a special kind of all previous kinds of regular groups).


  Repeating groups constitute clusters of 3 kinds: levels, pairs, and foursomes.

  Flats having the same gaps                                                                         constitute levels.

  Uniform symmetriads , curls, stairs , and zigzags,
               having vertical symmetry each to other,                                         constitute

  Uniform repeating irregular groups
               having both vertical and horizontal symmetry each to other           do

in the range until 1,000,000,000

V = 0
have 2, 3, 4, and 5 gaps
The consecutive equal gaps are repeating elements,
and the size is the number of consecutive gaps
Maximal size = 5
Gn G0

Vi = Vi+1
have 2, 3, 4, 5, 6, 7, 8, and 9 variations

Maximal size = 9

Vi = - Vi+1
have 3, 4, 5, and 6 variations

Maximal size = 6

Gc - i = Gc + i

 Vc - i = - Vc + i- 1
have 2, 4, 6, 8, 10,

Vc  = 0
c - i = - Vc + i
have 3, 5, and 7 variations
Maximal size = 7
Gc G0GcGn
G0      GVn V 0G0Gc G0



Gc - Gc - i = Gc + 1 - Gc
Vc - i =  Vc + i-1

have 4, 6, 8, and 10 variations

Maximal size = 10


Vc - i =  Vc + i
have 3, 5, 7,  9, 11, and 13   variations

Maximal size = 13


Irregular repeating groups constituting a foursome have the same set of variations 
and are of four kinds:

    groups having the sequence of variations downward:
                   - positive groups (main kind);
                   - negative groups having the same variations with the reversed sign;

   antigroups having the reversed order of the sequence of variations (upward):
                   - positive antigroups having the same variations with the reversed sign;
                   - negative antigroups having the same variations with the same sign.

Negative group

Positive group

Negative antigroup

Positive antigroup

Maximal size of each type of regular or irregular repeating groups
depends on its dynamic characteristics .

Some maximal groups may be unique

Up ] Introduction of Variation ] [ Regular and Repeating Groups ] Groups in the Range until 100,000 ] Groups in the Range until 1,000,000 ] Key of Primes Structure ] Distribution of gaps and variations ] Boolean Algebra of Classes ] Consecutive primes ] Groups of primes in the infinite set ] Groups of primes in a limited range ] We have for each kind of regular and repeating groups ]

Last updated 03/27/2009
Copyright 2003 Michael Chassis. All rights reserved.