All Complete Structures – Family A

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Family A

This family includes all complete structures (up to a reasonable size and eccentricity), and in the second table many derived structures. Structures are either rectilinear solids (built of body centered cubic arrangements of diquarks and monoquarks) or octahedra whose cores are such solids. Dirac gives a theoretic limit to z (charge on a nucleus when in an atom) of about 137, values of z past 133 can be ignored for real world situations (as the inner electrons crash into the nucleus destroying it during normal 1s orbits). Any of the 3 color schemes is possible for a single nucleus, but a nucleus can’t have monoquarks of 2 different colors. For simplicity only examples with red monoquarks are shown.

Eccentric solutions are marked with an E. Skew solutions with T for thin (starting from a 2×3 block less 2 corners) , an S for standard (starting from a 3×3 square less 2 corners) , H for half fat (starting from a 3×4 block less 2 corners) , or F for fat (starting from a 4×4 square less 2 corners) . The first of each of these patterns is the first degree instance, higher degree instances ad nauseum can be generated. Thin skew has equivalent solutions from other patterns, so is not deeply examined (since in each case the bond count distribution is worse than its twin). Skewing can effect 2 vertices, 4 vertices, or all 6 vertices, based on the number of planes skewed. Different vertices can be of different types (a fatly skewed vertex and a half-fat skewed vertex for instance), but not all combinations  are possible. Additionally, there are the Diagonal types D (with 3 spheres at each end) and DC (with intact 2x2x2 cubes at each end) occupying a diagonal through the center of a cube at the lowest layer (layer 1 of each  of these is a line of disconnected spheres), and diagonals based on other blocks (DT for thin 1x2x2 stacked diagonally, FDC for fat diagonal cubes, HDC for half-fat diagonal cubes), and diagonal types derived from the  symmetric and near symmetric solutions. See individual pages for details on each type.

Eccentric solutions have rectangular vertices (either 4 or 6 rectangular, depending on number of dimensions of eccentricity), skew solutions have diagonal vertices (2 per dimension of skewing). Each skew solution can be extended so all surface spheres with 5 neighbors gets a sixth, but that sphere is weakly bound to the structure, so non-extended are stronger. Extended solutions get an X suffix before the layer. All complete  structures have 6 total vertices, those neither rectangular nor diagonal are square. A few solutions are both Skew and Eccentric – those have formats S1E-1 etc., each has 2 diagonal vertices and 4 rectangular. To keep dimensionality 1 digit, dimensions higher than 9 use A and B to mean 10 and 11 (So S00AL1 is a skew solution with 10 degrees of skewing in a single dimension, while S000LB is a symmetric solution with 11 layers). Eccentricity from the 2x2x2 base can be either positive or negative: E-1-1 has a core that is 1x1x2. E+1+3 has a core that is 2x3x5. There are a few symmetric groups of solutions (Marked E0,0): even with a core of 2x2x2, even with a core of 4x4x4 (typically with 2×2 vertices added to each face), and odd with a core of 1x1x1 (each of these have 6 square vertices). Odd with 3x3x3 or 5x5x5 could be generated, but like the 1x1x1 case they  have parity violations, so why bother? The first even symmetric solution is also known as S000 (skew of 0 in all 3 dimensions) or DC0 (a single cube). D0 is equivalent to the first odd symmetric case. To distinguish among these, an O suffix is added for the first odd, and a Q for the second even (quad). Eccentric solutions derived from quad usually are beyond the useful portion of the ridge system, but the first few are included.

Overall degree of skewing can be calculated as square root of the sum of the squares of the individual dimensional skewing – ie S236 is degree 7, just as S007 is. Similar measures for eccentric and hybrid types can also be used. All structures with skewing under 7, and most through 10 are listed (all through skewing of 12 were included in calculations). Some groups have parallel members of other groups (from some point onward they exactly match interior and surface counts), a see also comment is added for those.  Odd solutions (with 3 odd dimensions) have a parity violation, as do hybrid solutions with odd eccentricity, but both are included for completeness. The first table just gives the complete structures: nearly complete structures with a vertex trimmed or removed are also feasible, and may be the actual best fit for a baryon count. Trimming of skew  vertices normally is 6 spheres at a time (2 sets of to ), trimming of eccentric or even symmetric is normally 4 spheres (again 2 sets but now to ). This removes an even chunk from both vertices of a pair, maintaining reflective symmetry and keeping bond count distribution similar. Addition of 2 trios (or larger triangular clumps of spheres) to appropriate spots may also be possible, for example extending S000L1 to D1L2. These addition, when not explicitly in another group, are marked with an r (so E-1-1+2t2L3 has 1 such addition), in the second table. For some larger solutions additions of 6 spheres, 10 spheres, or 15 spheres (triangles of 3, 4, and 5 on an edge) are possible. These are called sextets, dectets, and pents respectively. These are abbreviated t3, t4, t5, t6, …, tn, … . These overlays are typically are 2 less than the lowest edge count of a face, but can excede the face size (see DS001 for instance). If the triangle is not the largest allowed, adjacent spheres can be added covering 1 less sphere than is added (so adding 3 spheres at the edge of a dectet hides 2 spheres, growing the surface by 1 sphere). Solutions with 2 equivalent names are equated with the abbreviation aka (also known as). For instance: S001+2t2L1 aka D1L2. A symmetric addition involves 2 faces at a time (to maintain reflective symmetry). As all 8 faces can receive additions, long strings of addition symbols are possible: S001x+2t4+2t2+2sL4 has a pair of dectets, a pair of trios, and 2 single spheres added to an extended skew solution at layer 4 with 1 degree of skewing in a single dimension. Complicated cases have annotations in the link popup. For especially large base cases, adding a trimmed triangle with 3 corners removed is often a better fit. These are called hexagons with h2 being a dectet less 3, and h3 being a 15 sphere triangle less 3. In principle h4 and above could be generated.

A solution (whether complete or derived) has an overall eccentricity that is comprised of skew, eccentricity, and “fluffiness”. Where the last is the difference between the actual surface and an ideal octahedra, as exemplified by the odd symmetric case. Even the even symmetric is slightly fluffy, since the vertices are 2×2 squares rather than a single sphere. Cases where the solution is derived from the even symmetric with additions on all 8 faces have more surface per volume than the even symmetric itself, while still looking symmetric (even spherical) to the naked eye. For example, Fm-252 is derived from Po-224 with 8 h2’s added, increasing the surface by 32 spheres, but only increasing the total volume by 56 spheres.

Ridgelines of solutions occur with decreasing frequency (that is, with increasing length between final solutions). The “Best Fit” column is the active ridgeline, and “Nearly Best” is the ridgeline directly above (using the ratio of surface to interior, here normalized by adding 8 to each to prevent division by zero), which becomes best fit after the current ridge is exhausted. The final 2 solutions on a ridgeline are the best for the ridge and include the symmetric cases (odd on the odd ridgeline and even on the even ridgelines). “Other” solutions are beyond the nearly best (that is, they have more surface for the baryon count than either best fit or nearly best do). The best fit and nearly best for the first 18 ridgelines are grouped in family BE (even ridgelines), and family BO (odd). A shorter form of this family (A), with only best fit and nearly best that are reasonable z values (not beyond 150, which is slightly above Dirac’s limit of 137) can be found as FamilyG. All the solutions through a reasonable eccentricity on the first 22 ridgelines (and a bunch above those) are included here. Ridge 18 crosses Dirac’s limit in its best fit section, all ridges beyond that are above it (hence impossible) for the best fit portion of their run. The step up to the next ridgeline (both in surface size and degree of skewing) increases with overall size, but even the step up from ridge 22 to ridge 23 is only 20 surface spheres, and 5 skew degrees. The minimum ratio for the ridgeline (at the final solution on the ridgeline) is shown in  bold. Ridges of solutions are clean in the best fit section (essentially straight lines), but get wider away from the end point. Beyond the nearly best section they often overlap the ridges on either side.

 

 

Each cell has cell title plus surface sphere count, interior sphere count, baryon count (half the sum of above 2), ridge number, and often the count of complete cubes. Those solutions with at least 1 picture are marked Pic. Those rows or cells with a parity violation are highlighted in pink. Trimmed and extended solutions with asymmetric addition/subtraction are highlighted in blue. They probably are invalid, as they lack reflective symmetry. The approximate position within each ridgeline a group is found (the number of complete structures beyond it that exist within each ridge) is shown on the row title cell. For higher degrees of skewing mild variation in position is common. Exact position is calculated for best fit and nearly best. Parallel structures are considered as using only 1 position slot. In general, distance from the end of the ridge is proportionate to the degree of skewing. Where degree of skewing is similar, solutions may overlap.

 

In addition to surface versus interior, a particular solution has Dipole. This is the asymmetry in charge, and determines which reasonable size solution is actual best. For small nuclei, with many surface downs, dipole is an afterthought. For large solutions, where every surface monoquark is an up, it  becomes paramount. Solutions with high dipole (requiring more than 8 surface downs are highlighted in yellow, those requring more than 4, but fewer than 9 are in light yellow). Not all dipoles have been calculated so far, a white background does not guarantee low dipole. A computer analysis of dipole is neccessary to get an exact best match, especially where a solution has many variants. For instance S011+2t2+2sL6 has 2 trios of spheres and 2 single spheres added to its surface. These can be placed in trillions of places, hundreds of which are allowed by reflective symmetry, but 1 particular solution is the actual “best” for Astatine 220, leading to its 4 minute half-life. The others are no better than transient meta-states. Cases where 2 solutions have the same surface and interior may lead to longer lasting meta-states. An additional complication results from the existence of multiple charge surface, sets of spheres equidistant from the center of the nucleus. Solutions with few charge surfaces have less energy than otherwise equivalent surface with many, so a solution derived from the symmetric or near-symmetric solutions will be preferred over an otherwise similar solution.Click here to view the table.