Abstract
Paper 59 showed that the volume of the unit ball is greatest in $5$ dimensions, and Paper 64 concluded that no single magic dimension exists. This paper asks which dimension the packing question picks——the answer is $8$ and $24$. No new mathematical theorem and no new law is claimed.
First, the condition is a single angle. The centres of touching spheres must be at least $60$ degrees apart as seen from the middle (Section 2).
Second, this is the core of the paper. The area count allows $\mathbf{14}$, and only $\mathbf{12}$ fit (Section 3).
Third, there is room and still no thirteenth. Twelve on an icosahedron are $\mathbf{63.434949}$ degrees apart, $3.434949$ degrees more than needed (Section 4).
Fourth, settling it took $259$ years. Newton and Gregory disagreed in $1694$; it closed in $1953$ (Section 4).
Fifth, only six dimensions are known exactly. $1$, $2$, $3$, $4$, $8$ and $24$, and $8\to24$ jumps by $819$ (Section 5).
Sixth, the separator is what is being maximised. Volume picks $5$, closed orbits pick $3$, packing picks $8$ and $24$ (Section 6).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——kissing numbers, packing densities, the $E_8$ and Leech lattices, the Schutte--van der Waerden settlement and Viazovska’s proof are all standard. We do not build geometry——all we use is one angle and one ratio of areas. We do not prove the kissing number——the proof that $12$ is the maximum is not entered. That $12$ fit is checked here by construction; that $13$ do not is quoted. We do not build lattices——the constructions of $E_8$ and Leech are not entered. We write no number for the unsettled dimensions——for $d\ge5$ (apart from $8$ and $24$) the upper and lower bounds do not meet, and no figure is given. We do not enter coding theory——the correspondence between packings and error-correcting codes is not treated. Relation to earlier papers: Paper 59 showed the unit ball’s volume peaks in $5$ dimensions——this paper asks the same “which dimension is special” with a different question, and the answer is $8$ and $24$, not $5$. Paper 64 concluded that no single magic dimension exists——this paper adds one more independent instance to that conclusion. Paper 69 read $3+1$ as an over-determination by independent roots——packing does not join that over-determination, because it does not pick $3$. Paper 182 showed the curse of dimensionality is not one curse——this paper shows the gifts of dimension are likewise several. Paper 279 showed the cost of self-avoidance vanishing above four dimensions——it runs the same way as this paper: high dimensions do not simply get easier. What is added is computing that the area count allows $14$, measuring the icosahedral configuration at $63.434949$ degrees and the slack at $3.434949$, listing the six dimensions that are settled, and setting volume, closed orbits and packing side by side to show they pick different dimensions.
1. Notation and the Question
Setting: Place one sphere, then place equal spheres touching it. How many fit——this count is the kissing number.
Observation 1.1 Paper 59 showed that the unit ball’s volume peaks in $5$ dimensions, and Paper 64 concluded that no single magic dimension exists. Paper 69 read $3+1$ as an over-determination by independent roots.
Question: Which dimension, then, does the packing question pick?
Observation 1.2 The answer is $8$ and $24$. Neither the $5$ of volume nor the $3$ of orbits——change the question and the special dimension changes.
2. The Condition Is a Single Angle
Check 2.1: Put a unit sphere at the centre. A touching sphere’s centre is at distance $2$, and two neighbours’ centres must also be at least $2$ apart or they overlap.
Observation 2.2 Two sides of $2$ with an opposite side of at least $2$——the cosine rule gives an angular separation of at least $\mathbf{60}$ degrees as seen from the centre.
| Quantity | Value |
|---|---|
| Distance to a neighbour’s centre | $2$ |
| Minimum distance between neighbours | $2$ |
| Minimum angular separation required | $\mathbf{60.000000}$ degrees |
Observation 2.3 The problem has turned into one about points on a sphere. Radius and volume have gone, and only an angle is left——the same place where Paper 31 wrote that $\pi$ carries no dimension: the scale has dropped out.
3. The Core: The Area Count Allows $14$ and Only $12$ Fit
Setting: Each neighbour occupies a cap of angular radius $30$ degrees on the middle sphere (half of the $60$). Count how many such caps the sphere could hold.
Check 3.1:
| Quantity | Value |
|---|---|
| Cap area $/$ sphere area $=(1-\cos30°)/2$ | $0.06698730$ |
| Its reciprocal | $14.928203$ |
| Count allowed by area | $\mathbf{14}$ |
| Count that actually fits ($d=3$) | $\mathbf{12}$ |
Observation 3.2 This is the core of the paper. The area count allows $\mathbf{2}$ too many——because caps cannot tile a sphere without gaps.
Observation 3.3 That gap of $2$ is the difficulty, measured (Section 4). With an upper count of $14$ and a truth of $12$, an argument is needed that rules out $13$——area does not reach it.
Observation 3.4 Paper 224 wrote “flat wherever you touch it, and four pi in total.”Here too the local count and the global count disagree——adding areas does not settle whether they fit.
4. There Is Room, and Still No Thirteenth
Setting: There is more than one way to place $12$. Compare the cuboctahedron (vertices at the permutations of $(\pm1,\pm1,0)$) with the icosahedron.
Check 4.1: Computing the minimum separation.
| Arrangement | Points | Minimum separation |
|---|---|---|
| Cuboctahedron | $12$ | $\mathbf{60.000000}$ degrees (just touching) |
| Icosahedron | $12$ | $\mathbf{63.434949}$ degrees |
| Required minimum | — | $60$ degrees |
Observation 4.2 The icosahedral twelve have $\mathbf{3.434949}$ degrees to spare. None of them touches a neighbour and every one of them can be moved——the room is visibly there.
Observation 4.3 And still a thirteenth does not fit. The room exists but cannot be gathered into one place——existing and being usable are different.
Check 4.4: The history.
| Year | Event |
|---|---|
| $1694$ | Newton said $12$, Gregory thought $13$ |
| $1953$ | Schutte and van der Waerden settled it at $12$ |
| Gap | $\mathbf{259}$ years |
Observation 4.5 Gregory erred because he saw the room. The $3.434949$ degrees of slack are real——he mistook one thing only: that visible room need not be usable room.
Observation 4.6 A point to write honestly——this paper does not prove that $13$ cannot fit. That $12$ fit was checked here from the coordinates above——the upper bound is quoted.
5. Only Six Dimensions Are Known Exactly
Check 5.1:
| $d$ | Kissing number | Ratio to the row above |
|---|---|---|
| $1$ | $2$ | — |
| $2$ | $6$ | $3.0000$ |
| $3$ | $12$ | $2.0000$ |
| $4$ | $24$ | $2.0000$ |
| $8$ | $\mathbf{240}$ | $10.0000$ |
| $24$ | $\mathbf{196560}$ | $\mathbf{819.0000}$ |
Observation 5.2 Outside these six the bounds do not meet. Even $d=5$ is not settled——the ground next to $3$ is empty.
Check 5.3: The packing densities that are proved.
| $d$ | Density | Settled |
|---|---|---|
| $1$ | $1.0000000000$ | trivial |
| $2$ | $0.9068996821$ | Thue $1910$ |
| $3$ | $0.7404804897$ | Hales $1998$ |
| $8$ | $0.2536695079$ | Viazovska $2016$ |
| $24$ | $0.0019295743$ | Cohn et al. $2016$ |
Observation 5.4 The same list——$1$, $2$, $3$, $8$, $24$. The dimensions where the kissing number is known and where the density is known nearly coincide——not by accident but because the same lattices ($E_8$ and Leech) supply both answers.
Observation 5.5 Density falls with dimension. At $d=24$ it is $0.0019295743$——high-dimensional space is almost entirely gap. The same phenomenon Paper 182 counted as the curse, seen from another side.
6. Separator: What Is Being Maximised
Observation 6.1 Separator——which quantity the question asks to be greatest.
| Question | Dimension picked | Source |
|---|---|---|
| Volume of the unit ball is greatest | $\mathbf{5}$ | Paper 59 |
| Orbits close, waves travel sharply | $\mathbf{3}$ | Papers 64, 69 |
| $4\pi=2\pi\chi$ holds | $\mathbf{3}$ | Paper 128 |
| The cost of self-avoidance vanishes | $\mathbf{4}$ and above | Paper 279 |
| Spheres pack specially | $\mathbf{8}$ and $\mathbf{24}$ | this paper |
Observation 6.2 Five questions return four different answers. Paper 64’s conclusion that no single magic dimension exists gains one more independent instance——and this answer overlaps none of the others.
Observation 6.3 Applying Paper 300’s criteria.
| Criterion | Result |
|---|---|
| Is there a correspondence preserving identities? | no. Nothing links volume to packing |
| Does the agreement persist under motion? | no. Move $d$ and the answers part |
| Do the same quantities appear? | no. An integral against a lattice |
Observation 6.4 All three say no——“special dimension” is one phrase with several distinct roots. The same form as Paper 16’s six roots under “$=$” and Paper 313’s four under “synchrony”.
Observation 6.5 One last thing——$8$ and $24$ are special because special lattices live there. $E_8$ and Leech——it is not the dimension that is special but the object able to live in it. Not mistaking the subject is the discipline here.
7. Roll Call and Closing
- $60$ degrees is required (Check 2.1): Radius and volume drop out.
- Area allows $14$ (Check 3.1): The reciprocal of $0.06698730$.
- Only $12$ fit (Check 3.1): Two too many.
- The cuboctahedron sits at $60$ (Check 4.1): Just touching.
- The icosahedron at $63.434949$ (Check 4.1): $3.434949$ to spare.
- And no thirteenth (Observation 4.3): Existing is not being usable.
- $259$ years to settle (Check 4.4): $1694$ to $1953$.
- Six dimensions known (Check 5.1): $8\to24$ jumps by $819$.
- Density gives the same list (Check 5.3): $E_8$ and Leech supply both.
- Five questions, four answers (Observation 6.1): No single special dimension.
Closing: Paper 59 showed the unit ball’s volume peaks in $5$ dimensions and Paper 64 concluded that no single magic dimension exists. The packing question picks $8$ and $24$——neither the $5$ of volume nor the $3$ of orbits. The condition is a single angle——neighbouring centres must be at least $60$ degrees apart, and at that point radius and volume drop out and the problem becomes one about points on a sphere. And the area count allows $\mathbf{14}$ while only $\mathbf{12}$ fit——caps cannot tile a sphere, and that gap of $2$ is the difficulty itself. Twelve placed on an icosahedron are $\mathbf{63.434949}$ degrees apart, $\mathbf{3.434949}$ degrees more than the $60$ required——none of them touches, every one can be moved, and still a thirteenth does not fit. Gregory erred because he saw the room——the $1694$ dispute closed in $1953$, $\mathbf{259}$ years later. Only six dimensions are settled——$1$, $2$, $3$, $4$, $8$, $24$, with a factor of $\mathbf{819}$ from $8$ to $24$; the proved densities give nearly the same list. One thing separates them——which quantity the question asks to be greatest. Volume says $5$, closed orbits say $3$, $4\pi=2\pi\chi$ says $3$, self-avoidance says $4$ and above, packing says $8$ and $24$——five questions, four answers. One last thing——$8$ and $24$ are special because $E_8$ and the Leech lattice live there. It is not the dimension that is special but the object able to live in it, and the subject must not be mistaken.
References
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Reference
Yamagishi, Y. (2026). Which Dimension Is Special Depends on the Question [Paper 315]. YOI: https://www.yamagishiyuuki.com/yoi/315
DOI: 10.5281/zenodo.22208139
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 315. This paper is a preprint and has not been peer reviewed.