Kissing Number & PackingStar

A Visual Story

How a team of young researchers used AI to break 50-year-old records
— and what it reveals about the creative loop between humans and machines.

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Sources: arXiv:2511.13391 · PKU WeChat video · PackingStar GitHub

A 330-year-old question

In 1694, Newton asked: how many identical balls can touch one central ball without overlapping?

Newton said 12. Gregory said 13. Newton was right — but it took until 1953 to prove it.

The same question extends to higher dimensions: 4D, 5D, … 31D. In most dimensions, we still don't know the answer.

Kissing number K(n) = the maximum number of non-overlapping unit spheres that can all "kiss" (touch) one central unit sphere in n-dimensional space.

Why does it matter?

Known kissing numbers and geometric structures

From the PKU video: known kissing numbers (left) and geometric structures in different dimensions (right).

It connects to information theory: packing the most data into the fewest bits. Error-correcting codes, telecommunications, data storage — all depend on how efficiently you can "pack" signals in high-dimensional space.

In the team's words: "Using the fewest bits to compress the most information — that is the kissing number problem."

The scale of the challenge

Known kissing numbers from 7D to 15D — growing exponentially. Note 1932 (14D) and 2564 (15D).

As dimension increases, the numbers grow fast: 126 → 240 → 306 → … → 1932 → 2564. And the difficulty explodes — in 50 years, only about 7 advances, each using a completely different mathematical method.

"Even the best mathematicians felt helpless." — PKU video

The breakthrough idea: cosine matrix

The team tried placing balls by coordinates. It worked up to 8D, then numerical errors accumulated and destroyed everything.

Key insight: Instead of tracking coordinates in high-dimensional space, track only the cosine of the angle between each pair of sphere centres. This turns a geometry problem into a matrix-filling problem — numerically stable, GPU-friendly, and scalable.

In the PKU video, the researcher explained: "We turned a fill-in-the-blank exam into a multiple-choice test. Each cosine value becomes a choice from a clean, finite set."

Step 1: One player fills the matrix

The Filler agent selects cosine values from an action space to fill the Gram matrix.

The AI "plays a game": at each step, pick a cosine value and fill one entry in the matrix. When no more valid entries can be added, the game ends. Reward = how big the matrix got (= how many spheres fit).

This worked — but got stuck around 8–9 dimensions. The system couldn't scale to the really hard cases.

The 14D nightmare

The known kissing number in 14D is 1932. The team could not reproduce it. Thousands of experiments over six months — all failed.

"This 14D 1932 was like a nightmare, haunting us for over half a year. I even dreamed of reaching 1700+, but they were all illusions." — 马成栋 (Ma Chengdong)

Then a mathematician on the team noticed something: when he manually deleted a few bad balls, the system could suddenly recover the correct answer. Some balls were "spoiling the soup" — once added, they destroyed the whole structure.

Human insight: "Can we create a correction mechanism — a second agent that simulates the human manual correction?"

Step 2: Two players — the key innovation

Filler (+) adds entries; Corrector (−) removes bad ones. They cooperate through the game.

Filler (+)

Adds cosine values to grow the matrix. Tries to fit more spheres.

Corrector (−)

Removes "poisonous" entries. Inspired by the mathematician's instinct to delete bad balls by hand.

Once implemented, the system immediately solved the 14D problem, reaching 1932. A human insight, encoded into AI architecture, unlocked everything.

Scaling up: engineering matters

Without checkpoints: crash = restart from zero. With checkpoints: recover and keep going.

To reach 30+ dimensions, the team rewrote GPU kernels, designed mixed-precision arithmetic, and built a checkpoint system for uninterrupted deep search over days.

"Computation and intelligence are inseparable. If we can search faster, we are effectively smarter." — 程远 (Yuan Cheng)

Step 3: Fragment expansion — "cell division"

Each completed matrix is decomposed into fragments, which are recombined to seed new games.

After each game, the matrix is broken into geometric "fragments" — smaller sub-structures with good properties. These are recombined as seeds for new games.

"Like cell division, constantly producing new digital life." — PKU video

It was a strange fragment from the 200th experiment that led to the breakthrough in 25 dimensions.

The results: records shattered

Red stars = PackingStar's new records. Grey = previous best. Inset = improvement ΔK per dimension.

25D–31D: New lower bounds in all 7 dimensions. Improvement grows from +8 to +5,476.
13D: First rational advance since 1971 (1130 → 1146, breaking a 50-year record).
6,000+ new configurations discovered across multiple dimensions.
All results independently verified and accepted into official databases.

Visualisation: scale of improvement

Left: side-by-side comparison. Right: ΔK grows dramatically — from +8 (25D) to +5,476 (31D).

The exponential growth of improvements suggests PackingStar found fundamentally better construction strategies, not just incremental tweaks.

Visualisation: hidden algebraic order

Left: 12D has only 3 distinct cosine values. Right: 13D rational has exactly 6 rational fractions.

The AI didn't find random arrangements — it found configurations with extraordinary algebraic regularity. The 12D configuration (81 spheres) has every pair of spheres at one of just three angles. The 13D rational has exactly six values, all exact fractions: {−1, −1/2, −1/4, 0, 1/4, 1/2}.

Visualisation: structure inside the matrix

Block-diagonal patterns in the 12D Gram matrix reveal internal sub-structures ("fragments").

These patterns correspond to the geometric building blocks PackingStar discovers and reuses — the "fragments" from the cell-division process.

Visualisation: every sphere has exactly 36 contacts

12D contact graph: 81 nodes, 1,458 edges. Every node has degree exactly 36 — a strongly regular graph.

This perfect regularity is a hallmark of deep mathematical structure — the kind that connects to finite groups and combinatorial designs. The AI found it without being told to look for it.

AI overturned human assumptions

For decades, mathematicians assumed the best sphere packings would be antipodal — every ball has a partner on the opposite side. PackingStar found something unexpected:

"Several of the newly discovered configurations fundamentally challenge long-held antipodal paradigms. Rather than conforming to symmetry-driven constructions, these configurations exhibit asymmetric yet highly regular geometric patterns." — Paper

The AI showed mathematicians they were wrong about a decades-old assumption.

Then humans learned from the AI

This is what makes the story non-trivial. It's not just "AI finds, humans check." The loop goes both ways:

"These structural insights have, in turn, inspired mathematicians to extract new and more general construction principles. Building on patterns uncovered by PackingStar, subsequent human-designed configurations have further improved existing lower bounds in several dimensions, including 22 dimensions, illustrating a productive feedback loop between AI-driven discovery and mathematical theory." — Paper

And the hardest work? Not running the machine — interpreting what it found:

"The work I spent the most time on was interpreting the data from the AI black box. In ancient Greek there's a word for my role: 解释神 — interpreter of the gods." — PKU video

The creative loop

This is not the familiar story of "AI as a faster calculator checked by humans."

Step	Who	What happened
1	Human	Posed the problem (Newton, Hilbert, centuries of mathematicians)
2	Human	Designed the cosine matrix formulation and two-player game
3	Human	Observed "delete bad balls" instinct → created Corrector agent
4	AI	Discovered configurations that overturned antipodal assumptions
5	Human	Interpreted surprises, extracted new construction principles
6	Human	Used those principles to improve bounds in other dimensions

Each side transformed the other's understanding.

More from the data

Eigenvalues confirm rank = dimension (12 and 13).

17D → 3D projection shows layered shells.

Triangle building blocks for 29D and 31D records.

13D: exactly 6 rational cosine values.

"Human and machine exploring the universe together"

"This road felt dark and endless. But after the breakthrough, it was as if the sky suddenly brightened — and a wider starfield was waiting for us. PackingStar is the spacecraft carrying us to explore that starfield. This is not an arrival, but a brand-new beginning."
— PKU video closing

The work was done by six young researchers — cross-disciplinary, combining mathematics and AI. As the team put it: "AI helps these young scientists explore the unknown universe more efficiently. That itself is a kind of romance between humans and machines."

AI maths romance is a two-way street

SCMP Letter Draft (~194 words)

I refer to your report on the Chinese team's use of AI to advance the "kissing number" problem ("Romance between scientists and AI," 19 February).

The kissing number asks how many identical balls can touch one central ball without overlapping—a question Newton posed in 1694, still unsolved in most dimensions. PackingStar's record-breaking results, including the first advance in 13 dimensions since 1971, deserve celebration.

Your headline calls this a "romance," and the metaphor runs deeper than it might seem. A key design idea—a second AI agent that removes flawed choices—came from watching a mathematician delete bad entries by hand: the team taught their machine to mimic human instinct. But the machine returned the favour, discovering sphere arrangements that overturned assumptions held for decades. Their hardest task was "interpreting the data from the AI black box"—reading meaning into surprises neither side could have produced alone.

That is what makes this a genuine romance, not mere human oversight of a fast calculator: each partner transforms the other. If we want AI-driven research to bear fruit, we should invest not just in algorithms, but in the human expertise to question, interpret, and build on what AI reveals.

AI maths romance is a two-way street

Final Letter — submitted to SCMP

Simon Wang

I refer to your report on the Chinese team's use of AI to advance the "kissing number" problem ("Romance between scientists and AI," 19 February).

~194 words

How AI helped me write this letter

A transparent account of the human–AI collaboration behind this letter.

What I (the human) brought

The hook: I chose the SCMP report and decided this was worth a letter.
The angle: I proposed "human in the loop" as the core argument, based on my reading of the paper and the PKU video.
The sharpening: When the first draft sounded like a platitude ("of course humans matter"), I pushed back — "readers will say they don't need me to tell them that." This forced a reframe from supervision to creative loop.
Source selection: I pointed the AI to the paper, the PKU WeChat video, the SCMP report, and the GitHub repo.
Quality control: I reviewed each draft, checked claims against sources, and decided what to keep or cut.

What AI contributed

Research at scale: AI read the full paper (HTML), the PKU video transcript (Chinese), and the GitHub codebase, extracting specific quotes and evidence I could not have processed as quickly.
Drafting iterations: AI produced four drafts (firstDraft → draft01 → draft02 → draft03), each responding to my feedback on thesis, tone, jargon, and word count.
Source-checking: AI traced each claim in the letter to an exact quote and location (paper paragraph, video timestamp).
Visualisations: AI generated 8 data visualisations from PackingStar's raw data, which deepened my understanding of what the AI system actually found.
Explainer deck: AI built this presentation (explain01–04.html) to help me understand the problem from scratch — essential since I have no advanced maths background.

The process, step by step

Step	Who led	What happened
1	Human	Found the SCMP report; decided to write a letter
2	Human	Pointed AI to the arXiv paper, PKU video, SCMP report
3	AI	Located paper on arXiv, downloaded, extracted key findings
4	Human	Proposed "human in the loop" angle; asked for brainstorming
5	AI	Created ideas4letters.md, moreIdeas.md from paper + video
6	AI	Built explainer decks (explain01–02) to teach me the problem
7	AI	Wrote firstDraft.md (~148 words)
8	Human	Feedback: stronger thesis, less jargon, more celebration
9	AI	Wrote draft01.md (~199 words) with revised thesis
10	Human	"This sounds trivial — readers will say 'of course humans matter'"
11	AI	Re-read all sources; reframed from platitude to "creative loop"
12	AI	Wrote draft02.md (~186 words) with sourced evidence
13	Human	Approved draft; requested exact source quotes for each claim
14	AI	Added sentence-by-sentence source annotations (paper + video)
15	Human	Chose title; cloned GitHub repo; asked AI for 8 data visualisations
16	AI	Built explain03–04.html with visualisations, keyframe images, full narrative
17	Human	Last-minute insight: the SCMP "romance" metaphor is structurally accurate — AI mimics human instinct, humans learn from AI surprises. Rewrote the angle.
18	AI	Wrote draft03.md (~194 words) pivoting on "romance" with new title
19	Human	Final review, fact-check, submission

The parallel: Just as PackingStar's breakthrough came from a romance between AI and mathematicians, this letter itself emerged from the same kind of two-way street. The human set direction, pushed back on clichés, and — at the last minute — spotted that the SCMP's "romance" metaphor was the key to the whole argument. The AI processed sources at scale, iterated through four drafts, and handled research grunt work. Each partner transformed the other's thinking. Neither could have produced this alone.