In some respects, the combinations of motions with greater rotational displacement, those which constitute the atoms of the chemical elements, are less complicated than those with the least displacement, the sub-atomic particles, and it will therefore be convenient to discuss the structure of these larger units first.

Geometrical considerations indicate that two photons can rotate around the same central point without interference if the rotational speeds are the same, thus forming a double unit. The nature of this combination can be illustrated by two cardboard disks interpenetrated along a common diameter C. The diameter A perpendicular to C in disk *a* represents one linear oscillation, and the disk *a is* the figure generated by a one-dimensional rotation of this oscillation around an axis *B* perpendicular to both A and C. Rotation of a second linear oscillation, represented by the diameter * B*. around axis A generates the disk *b*. It is then evident that disk *a* may be given a second rotation around axis A, and disk *b* may be given a second rotation around axis *B* without interference at any point, as long as the rotational speeds are equal.

The validity of the mathematical principles of probability is covered in the fundamental postulates by specifically including them in the definition of “ordinary commutative mathematics,” as that term is used in the postulates. The most significant of these principles, so far as the atomic structures are concerned, are that small numbers are more probable than large numbers, and symmetrical combinations are more probable than asymmetrical combinations of the same total magnitude. For a given number of units of net rotational displacement the double rotating system results in lower individual displacement values, and the probability principles give them precedence over single units in which the individual displacements are higher. All rotating combinations with sufficient net total displacement to enable forming double units therefore do so.

To facilitate a description of these objects we will utilize a notation in the form *a-b-c*, where c is the speed displacement of the one-dimensional reverse rotation, and a and *b* are the displacements in the two dimensions of the basic two-dimensional rotation. Later in the development we will find that the one-dimensional rotation is connected with electrical phenomena, and the two-dimensional rotation is similarly connected with magnetic phenomena. In dealing with the atomic and particle rotations it will be convenient to use the terms “electric” and “magnetic” instead of “one-dimensional” and “two-dimensional” respectively, except in those cases where it is desired to lay special emphasis on the number of dimensions involved. It should be understood, however, that designation of these rotations as electric and magnetic does not indicate the presence of any electric or magnetic forces in the structures now being described. This terminology has been adopted because it not only serves our present purposes, but also sets the stage for the introduction of electric and magnetic phenomena in a later phase of the development.

Where the displacements in the two magnetic dimensions are unequal, the rotation is distributed in the form of a spheroid. In such cases the rotation which is effective in two dimensions of the spheroid will be called the *principal* magnetic rotation, and the other the *subordinate* magnetic rotation. When it is desired to distinguish between the larger and the smaller magnetic rotational displacements, the terms *primary* and * secondary will* be used. Where motion in time occurs in the material structures now being discussed, the negative displacement values of this motion will be distinguished by placing them in parentheses. All values not so identified refer to positive displacement (motion in space).

Some questions now arise as to the units in which the displacements should be expressed. As will quickly be seen when we start to identify the individual structures, the natural unit of displacement is not a convenient unit in application to the double rotating systems. The smallest change that can take place in these systems involves two natural units. As stated in Chapter 9, probability considerations dictate the distribution of the total displacement of a combination among the different dimensions of rotation. The possible rotating combinations therefore constitute a series, successive members of which differ by two of the natural one-dimensional units of displacement. Since we will not encounter single units in these atomic structures, it will simplify our calculations if we work with double units rather than the single natural units. We will therefore define the unit *of electric displacement* in the atomic structures as the equivalent of two natural one-dimensional displacement units.

On this basis, the position of each element in the series of combinations, as determined by its net total equivalent electric displacement, is its *atomic number*. For reasons that will be brought out later, half of the unit of atomic number has been taken as the unit of *atomic weight.*

At the unit level dimensional differences have no numerical effect; that is, 1^{3} = 1^{2} = 1. But where the rotation extends to greater displacement values a two-dimensional displacement n is equivalent to n^{2} one-dimensional units. If we let n represent the number of units of electric displacement, as defined above, the corresponding number of natural (single) units is 2n, and the natural unit equivalent of a magnetic (two-dimensional) displacement n is 4n^{2}, Inasmuch as we have defined the electric displacement unit as two natural units, it then follows that a magnetic displacement n is equivalent to 2n^{2} electric displacement units.

This means that the unit *of magnetic displacement*, the increment between successive values of the two-dimensional rotational displacement, is not a specific magnitude in terms of total displacement. Where the total displacement is the significant factor, as in the position in the series of elements, the magnetic displacement value must be converted to equivalent electric displacement units by means of the 2n^{2} relation. For some other purposes, however, the displacement value in terms of magnetic units does have significance in its own right, as we will see in the pages that follow.

In order to qualify as an atom—a double rotating system—a rotational combination must have at least one effective magnetic displacement unit in each system, or, expressing the same requirement in a different way, it must have at least one effective displacement unit in each of the magnetic dimensions of the combination structure. One positive magnetic (double) displacement unit is required to neutralize the two single negative displacement units of the basic photons; that is, to bring the total scalar speed of the combination as a whole down to zero (on the natural basis). This one positive unit is not part of the effective rotation. Thus, where there is no rotation in the electric dimension, the smallest combination of motions that can qualify as an atom is 2-1-0. This combination can be identified as the element helium, atomic number 2.

Helium is a member of a family of elements known as the *inert gases*, a name that has been applied because of their reluctance to enter into chemical combinations. The structural feature that is responsible for this chemical behavior is the absence of any effective rotation in the electric dimension. The next element of this type has one additional unit of magnetic displacement. Since the probability factors operate to keep the eccentricity at a minimum, the resulting combination is

2-2-0, rather than 3-1-0. The succeeding increments of displacement similarly go alternately to the principal and subordinate rotations.

Helium, 2-1-0, already has one effective displacement unit in each magnetic dimension, and the increase to 2-2-0 involves a second unit in one dimension. As previously indicated, the electric equivalent of n magnetic units is 2n^{2}. Unlike the addition of another electric unit, the addition of a magnetic unit is not a simple process of going from 1 to 2. In the case of the electric displacement there is first a single unit, then another single unit for a total of two, another bringing the total to three, and so on. But 2 × 1^{2} = 2, and 2 × 2^{2} = 8. In order to increase the total electric equivalent of the magnetic displacement from 2 to 8 it would be necessary to add the equivalent of 6 units of electric displacement, and there is no such thing as a magnetic equivalent of 6 electric units. The same situation arises in the subsequent additions, and the increase in magnetic displacement must therefore take place in full 2n^{2} equivalents. Thus the succession of inert gas elements is not 2, 10, 16, 26, 36, 50, 64, as it would be if 2(n+1)^{2} replaces 2n^{2} in the same manner that n+ l replaces n in the electric series, but 2, l0, 18, 36, 54, 86, 118. For reasons which will be developed later, element 118 is unstable, and disintegrates if formed. The preceding six members of this series constitute the inert gas family of elements.

The number of mathematically possible combinations of rotations is greatly increased when electric displacement is added to these magnetic combinations, but the combinations that can actually exist as elements are limited by probability considerations, as noted in Chapter 9. The magnetic displacement is numerically less than the equivalent electric displacement, and is more probable for this reason. Its status as the essential basic rotation also gives it precedence over the electric rotation. Any increment of displacement consequently adds to the magnetic rotation if possible, rather than to the rotation in the electric dimension. This means that the role of the electric displacement is confined to filling in the intervals between the inert gas elements.

On this basis, if all rotational displacement in the material system were positive, the series of elements would start at the lowest possible magnetic combination, helium, and the electric displacement would increase step by step until it reached a total of 2n^{2} units, at which point the relative probabilities would result in a conversion of these 2n^{2} electric units into one additional unit of magnetic displacement, whereupon the building up of the electric displacement would be resumed. This behavior is modified, however, by the fact that electric displacement in ordinary matter, unlike magnetic displacement, may be negative instead of positive.

The restrictions on the kinds of motions that can be combined do not apply to minor components of a system of motions of the same type, such as rotations. The net effective rotation of a *material* atom must be in space in order to give rise to those properties which are characteristic of ordinary matter. It necessarily follows that the magnetic displacement, which is the major component of the total, must be positive. But as long as the larger component is positive, the system as a whole can meet the requirement that the net rotation be in space (positive displacement) even if the smaller component, the electric displacement, is negative. It is possible, therefore, to increase the net positive displacement a given amount either by direct addition of the required number of positive electric units, or by adding a magnetic unit and then adjusting to the desired intermediate level by adding the appropriate number of negative units.

Which of these alternatives will actually prevail is affected to a considerable degree by the conditions that exist in the atomic environment, but in the absence of any bias due to these conditions, the determining factor is the size of the electric displacement, lower displacement values being more probable than higher values. In the first half of each group intermediate between two inert gas elements, the electric displacement is minimized if the increase in atomic number (equivalent electric displacement) is accomplished by direct addition of positive displacement. When n^{2} units have been added, the probabilities are nearly equal, and as the atomic number increases still further, the alternate arrangement becomes more probable. In the latter half of each group, therefore, the increase in atomic number is normally attained by adding one unit of magnetic displacement, and then reducing to the required net total by adding negative electric displacement, eliminating successive units of the latter to move up the atomic series.

By reason of the availability of negative electric displacement as a component of the atomic rotation, an element with a net displacement less than that of helium becomes possible. Adding one negative electric displacement unit to helium produces this element, 2-1-(1), which we identify as hydrogen,, and thereby, in effect, subtracting one positive electric unit from the equivalent of two units (above the rotational base) that helium possesses. Hydrogen is the first in the ascending series of elements, and we may therefore give it the atomic number 1. The atomic number of any other material element is its net equivalent electric displacement.

Above helium, 2-1-0, we find lithium, 2-1-1, beryllium, 2-1-2, boron, 2-1-3, and carbon, 2-1-4. Since this is an 8-atom group, the probabilities are nearly even at this point, and carbon can also exist as 2-2-(4). The elements that follow move up the atomic series by reducing the negative displacements: nitrogen, 2-2-(3), oxygen, 2-2-(2), fluorine, 2-2-(1), and finally the next inert gas, neon, 2-2-0.

Another similar 8-element group follows, adding a second magnetic unit in the other magnetic dimension. This carries the series up to another inert gas element, argon, 3-2-0. Table 1 shows the normal displacements of the elements to, and including, argon.

At element 18, argon, the magnetic displacement has reached a level of two units above the rotational base in each of the magnetic dimensions.

Displacements | Element | Atomic Number | Displacements | Element | Atomic Number |
---|---|---|---|---|---|

2-1-(1) | Hydrogen | 1 | |||

2-1-0 | Helium | 2 | |||

2-1-1 | Lithium | 3 | 2-2-1 | Sodium | 11 |

2-1-2 | Beryllium | 4 | 2-2-2 | Magnesium | 12 |

2-1-3 | Boron | 5 | 2-2-3 | Aluminum | 13 |

2-1-4 2-2-(4) | Carbon | 6 | 2-2-4 2-2-(4) | Silicon | 14 |

2-2-(3) | Nitrogen | 7 | 3-2-(3) | Phosphorus | 15 |

2-2-(2) | Oxygen | 8 | 3-2-(2) | Sulfur | 16 |

2-2-(1) | Fluorine | 9 | 3-2-(1) | Chlorine | 17 |

2-2-0 | Neon | 10 | 3-2-0 | Argon | 18 |

Displacements | Element | Atomic Number | Displacements | Element | Atomic Number |
---|---|---|---|---|---|

3-2-1 | Potassium | 19 | 3-3-1 | Rubidium | 37 |

3-2-2 | Calcium | 20 | 3-3-2 | Strontium | 38 |

3-2-3 | Scandium | 21 | 3-3-3 | Yttrium | 39 |

3-2-4 | Titanium | 22 | 3-3-4 | Zirconium | 40 |

3-2-5 | Vanadium | 23 | 3-3-5 | Niobium | 41 |

3-2-6 | Chromium | 24 | 3-3-6 | Molybdenum | 42 |

3-2-7 | Manganese | 25 | 3-3-7 | Technetium | 43 |

3-2-8 | Iron | 26 | 3-3-8 | Ruthenium | 44 |

3-2-9 3-3-(9) | Cobalt | 27 | 3-3-9 3-3-(9) | Rhodium | 45 |

3-3-(8) | Nickel | 28 | 4-3-(8) | Palladium | 46 |

3-3-(7) | Copper | 29 | 4-3-(7) | Silver | 47 |

3-3-(6) | Zinc | 30 | 4-3-(6) | Cadmium | 48 |

3-3-(5) | Gallium | 31 | 4-3-(5) | Indium | 49 |

3-3-(4) | Germanium | 32 | 4-3-(4) | Tin | 50 |

3-3-(3) | Arsenic | 33 | 4-3-(3) | Antimony | 51 |

3-3-(2) | Selenium | 34 | 4-3-(2) | Tellurium | 52 |

3-3-(1) | Bromine | 35 | 4-3-(1) | Iodine | 53 |

3-3-0 | Krypton | 36 | 4-3-0 | Xenon | 54 |

In order to increase the rotation in either dimension by an additional unit a total of 2×3^{2}, or 18, units of electric displacement are required. This results in a group of 18 elements which reaches the midpoint at cobalt, 3-2-9, and terminates at krypton, 3-3-0. A second 18-element group follows, as indicated in Table 2.

The final two groups of elements, Table 3, contain 2×4^{2}, or 32, members

Displacements | Element | Atomic Number | Displacements | Element | Atomic Number |
---|---|---|---|---|---|

4-3-1 | Cesium | 55 | 4-4-1 | Francium | 87 |

4-3-2 | Barium | 56 | 4-4-2 | Radium | 88 |

4-3-3 | Lanthanum | 57 | 4-4-3 | Actinium | 89 |

4-3-4 | Cerium | 58 | 4-4-4 | Thorium | 90 |

4-3-5 | Praseodymium | 59 | 4-4-5 | Protactinium | 91 |

4-3-6 | Neodymium | 60 | 4-4-6 | Uranium | 92 |

4-3-7 | Promethium | 61 | 4-4-7 | Neptunium | 93 |

4-3-8 | Samarium | 62 | 4-4-8 | Plutonium | 94 |

4-3-9 | Europium | 63 | 4-4-9 | Americium | 95 |

4-3-10 | Gadolinium | 64 | 4-4-10 | Curium | 96 |

4-3-11 | Terbium | 65 | 4-4-11 | Berkelium | 97 |

4-3-12 | Dysprosium | 66 | 4-4-12 | Californium | 98 |

4-3- 13 | Holmium | 67 | 4-4-13 | Einsteinium | 99 |

4-3-14 | Erbium | 68 | 4-4-14 | Fermium | 100 |

4-3-15 | Thulium | 69 | 4-4-15 | Mendelevium | 101 |

4-3-16 4-3-(16) | Ytterbium | 70 | 4-4-16 5-4-(16) | Nobelium | 102 |

4-4-(15) | Lutetium | 71 | 5-4-(15) | Lawrencium | 103 |

4 4-(14) | Hafnium | 72 | 5-4-(14) | Rutherfordium | 104 |

4 4-(13) | Tantalum | 73 | 5 4-(13) | Hafnium | 105 |

4 4-(12) | Tungsten | 74 | 5-4-(12) | Seaborgium | 106 |

4 4-(11) | Rhenium | 75 | 5 4-(11) | Bohrium | 107 |

4 4-(10) | Osmium | 76 | 5-4-(10) | Hassium | 108 |

4-4-(9) | Iridium | 77 | 5-4-(9) | Meitnerium | 109 |

4-4-(8) | Platinum | 78 | 5-4-(8) | Darmstadtium | 110 |

4-4-(7) | Gold | 79 | 5-4-(7) | Roentgenium | 111 |

4-4-(6) | Mercury | 80 | 5-4-(6) | 112 | |

4-4-(5) | Thallium | 81 | 5-4-(5) | 113 | |

4-4-(4) | Lead | 82 | 5-4-(4) | 114 | |

4-4-(3) | Bismuth | 83 | 5-4-(3) | 115 | |

4-4-(2) | Polonium | 84 | 5-4-(2) | 116 | |

4-4-(1) | Astatine | 85 | 5-4-(1) | 117 | |

4-4-0 | Radon | 86 |

each. The heaviest elements of the last group have not yet been observed, as they are highly radioactive, and consequently unstable in the terrestrial environment. In fact, uranium, element number 92, is the heaviest that exists naturally on earth in any substantial quantities. As we will see later, however, there are other conditions under which the elements are stable all the way up to number 117.

For convenience in the subsequent discussion these groups of elements will be identified by the magnetic n value, with the first and second groups in each pair being designated A and B respectively. Thus the sodium group, which is the second of the 8-element groups (n=2) will be called Group 2B.

At this point it will be appropriate to refer back to this statement that was made in Chapter 9:

The (mathematical) development will begin with nothing more than the series of cardinal numbers and the geometry of three dimensions. By subjecting these to simple mathematical processes, the applicability of which to the physical universe of motion is specified in the fundamental postulates, the combinations of rotational motions that can exist in the theoretical universe will be ascertained, and it will be shown that these rotational combinations that theoretically can exist can be individually identified with the atoms of the chemical elements and the sub-atomic particles that are observed to exist in the physical universe. A unique group of numbers representing the different rotational components will be derived for each of these combinations.

A review of the manner in which the figures presented in Tables 1 to 3 were derived will show that this commitment, so far as it applies to the elements, has been carried out in full. This is a very significant accomplishment. Both the *existence* of a series of theoretical elements identical with the observed series of chemical elements, and the *numerical values* which theoretically characterize each individual element have been derived from the *general properties* of mathematics and geometry, without making any supplementary assumptions or introducing any numerical values specifically applicable to matter. The possibility that the agreement between the series of elements thus derived and the known chemical elements could be accidental is negligible, and this derivation is, in itself, a conclusive proof that the atoms of matter are combinations of motions, as asserted by the Reciprocal System of theory. But this is only the beginning of a vast process of mathematical development. The numerical values at which we have arrived, the atomic numbers and the three displacement values for each element, now provide us with the basis from which we can derive the quantitative relations in the areas that we will examine.

The behavior characteristics, *or properties*, of the elements are functions of their respective displacements. Some are related to the total net effective displacement (equal to the atomic number in the combinations thus far discussed), some are related to the electric displacement, others to the magnetic displacement, while still others follow a more complex pattern. For instance, valence, or chemical combining power, is determined by * either* the electric displacement or one of the magnetic displacements, while the inter-atomic distance is affected by *both* the electric and magnetic displacements, but in different ways. The manner in which the magnitudes of these properties for specific elements and compounds can be calculated from the displacement values has been determined for many properties and for many classes of substances. These subjects will be considered individually in the chapters that follow.

One of the most significant advances toward an understanding of the relations between the structures of the different chemical elements and their properties was the development of the periodic table by Mendeleeff in l869. In this diagram the elements are arranged horizontally *in periods* and vertically in *groups*, the order within the period being that of the atomic number (approximately defined in the original work by the atomic weights). When the elements are correctly assigned in the periods, those in the vertical groups are quite similar in their properties. On comparing the periodic table with the rotational characteristics of the elements, as tabulated in this chapter, it is evident that the horizontal periods reflect the magnetic rotational displacement, while the vertical groups represent the electric rotational displacement. In revising the table to take advantage of the additional information derived from the Reciprocal System of theory we may therefore replace the usual group and period numbering by the more meaningful displacement values.

When this is done it is apparent that a further revision of the tabular arrangement is required in order to put all of the elements into their proper positions. Mendeleeff’s original table included nine vertical groups, beginning with the inert gases, Group *O*, and ending with a group in which the three elements iron, cobalt, and nickel, and the corresponding elements in the higher periods, were all assigned to a single vertical position. In the more modern versions of the table the number of vertical groups has been expanded to avoid splitting each of the longer periods into two sub-periods, as was done by Mendeleeff. One of the most popular of these revised versions utilizes 18 vertical groups, and puts 15 elements of each of the last two periods into one of these l8 positions in order to accommodate the full number of elements.

In the light of the new information now available, it can be seen that Mendeleeff based his arrangement on the relations existing in the 8-element rotational groups, 2A and 2B in the notation used in this work, and forced the elements of the larger groups into conformity with this 8-element pattern. The modern revisers have made a partial correction by setting up their tables on the basis of the l8-element rotational groups, 3A and 3B, leaving blank spaces where the 8-element groups have no counterparts of the l8-element values. But these tables still retain a part of the original distortion, as they force the members of the 32-element groups into the l8-element pattern. To construct a complete and accurate table, it is only necessary to carry the revision procedure one step farther, and set up the table on the basis of the largest of the magnetic groups, the 32-element groups 4A and 4B.

For some purposes a simple extension of the current versions of the table to the full 32 position width necessary to accommodate Groups 4A and 4B is probably all that is needed. On the other hand, the useful chemical information displayed by the table is confined mainly to the elements with electric displacements below l0, and separating the central elements of the two upper groups from the main portion of the table, as in the conventional arrangements, has considerable merit. The particular elements that are thus separated on the basis of the electric displacement are not the same ones that are treated separately in the conventional tables, but the general effect is much the same.

When the table is thus divided into two sections, it also appears that there are some advantages to be gained by a vertical, rather than a horizontal, arrangement, and the revised table, Figure 1, has been set up on this basis. The new concept of “divisions,” which is emphasized in this table, will be explained in Chapter 18. Inasmuch as carbon and silicon play both positive and negative roles rather freely, they have each been assigned to two positions in the table, but hydrogen, which is usually shown in two positions in the conventional tables, is necessarily negative on the basis of the principles that have been developed in this work and is only shown in one position. The aspects of its chemical behavior that have led to its classification with the electropositive elements will also be explained in Chapter 18.

In the original construction of the periodic table the known properties of certain elements were combined with the atomic number sequence to establish the existence of the relations between the elements of the various periods and groups, and thereby to predict previously undetermined properties, and even the existence of some previously unknown elements. The table thus added significantly to the chemical knowledge of the time. In this work, however, the revised table is not being presented

Magnetic Displacement | Div. | Electric Displacement | Div. | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

2-1 | 2-2 | 3-2 | 3-3 | 4-3 | 4-4 | ||||||

3 Li | 11 Na | 19 K | 37 Rb | 55 Cs | 87 Fr | I | 1 | ||||

4 Be | 12 Mg | 20 Ca | 38 Sr | 56 Ba | 88 Ra | 2 | II | 4-3 | 4-4 | ||

5 B | 13 Al | 21 Sc | 39 Y | 57 La | 89 Ac | 3 | 10 | 64 Gd | 96 Cm | ||

6 C | 14 Si | 22 Ti | 40 Zr | 58 Ce | 90 Th | 4 | 11 | 65 Tb | 97 Bk | ||

23 V | 41 Nb | 59 Pr | 91 Pa | II | 5 | 12 | 66 Dy | 98 Cf | |||

24 Cr | 42 Mo | 60 Nd | 92 U | 6 | 13 | 67 Ho | 99 Es | ||||

25 Mn | 43 Tc | 61 Pm | 93 Np | 7 | 14 | 68 Er | 90 Fm | ||||

26 Fe | 44 Ru | 62 Sm | 94 Pu | 8 | 15 | 69 Tm | 101 Md | ||||

27 Co | 45 Rh | 63 Eu | 95 Am | 9 | 16 | 70 Yb | 102 No | ||||

28 Ni | 46 Pd | 78 Pt | 110 | III | (8) | (15) | III | 71 Lu | 103 Lr | ||

29 Cu | 47 Ag | 79 Au | 111 | (7) | (14) | 72 Hf | 104 Rf | ||||

30 Zn | 48 Cd | 80 Hg | 112 | (6) | (13) | 73 Ta | 105 Ha | ||||

31 Ga | 49 In | 81 Tl | 113 | (5) | (12) | 74 W | 106 | ||||

6 C | 14 Si | 32 Ge | 50 Sn | 82 Pb | 114 | IV | (4) | (11) | 75 Re | 107 | |

7 N | 15 P | 33 As | 51 Sb | 83 Bi | 115 | (3) | (10) | 76 Os | 108 | ||

8 O | 16 S | 34 Se | 52 Te | 84 Po | 116 | (2) | (9) | 77 Ir | 109 | ||

1 H | 9 F | 17 Cl | 35 Br | 53 I | 85 At | 117 | (1) | 4-4 | 5-4 | ||

2 He | 10 Ne | 18 Ar | 36 Kr | 54 Xe | 86 Rn | ||||||

2-1 | 2-2 | 3-2 | 3-3 | 4-3 | 4-4 | 5-4 | 0 |

as an addition to the information contained in the preceding pages, but merely as a convenient graphic method of expressing some portions of that information. Everything that can be learned from the table has already been set forth in more detailed form, verbally and mathematically, in this and the earlier chapters. Some of the implications of this information, such as its application to the property of valence, will have further consideration later.

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