Discrete geometry and three-dimensional packing

Densest regular tetrahedron packing

Collaboration beta

How densely can congruent regular tetrahedra fill space? The source reports sharp bounds for several constrained arrangements around the double-dimer candidate, while the unrestricted optimum remains unresolved.

Δ (T) = \sup_{P} {limsup}_{L \to \infty} \frac{| Q_{L} \cap ⋃ P |}{L^{3}} \overset{?}{=} \frac{4000}{4671}

Known results and sources

Illustration: Congruent tetrahedra may turn freely, but their interiors must not overlap.

Research problem

Exact mathematical statement

Let $T\subset\mathbb R^3$ be a regular tetrahedron of volume $9$ . A packing is a locally finite family $\mathcal P=\{x_i+R_iT\}$ , with $x_i\in\mathbb R^3$ , $R_i\in SO(3)$ , and pairwise disjoint interiors. Face, edge, and vertex contact is permitted. With $Q_L=[-L/2,L/2]^3$ , determine

\Delta(T)=\sup_{\mathcal P}\overline\delta(\mathcal P),\qquad \overline\delta(\mathcal P)=\limsup_{L\to\infty} \frac{|Q_L\cap\bigcup\mathcal P|}{L^3}.

The current candidate is

\Delta(T)=\delta_*=\frac{4000}{4671}.

The quantifier includes all orientations and every periodic or nonperiodic arrangement. The source attributes the attaining double-dimer construction to Chen–Engel–Glotzer. A packing of greater density would refute this candidate constant but would not alone determine the new optimum.

Problem infographic

Problem at a glance

Illustration: A projected regular tetrahedron and three differently oriented copies accompany the nonoverlap rule and the open density target 4000/4671. — Illustration: The optimization ranges over every periodic or nonperiodic packing. The displayed solids are separated examples, not an extremal packing.

Current mathematical picture

Where work on Densest regular tetrahedron packing stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureUniversal lower bounds for raw centroid Voronoi cells or whole-face pairs

The exact cluster rules out a universal partition into singletons and whole-face dimers with a separate raw-cell lower bound v_* per tetrahedron. Compensated or grouped volume accounting must permit unequal reference cells and control all deficit regions.

Route status · Narrowed route

Main reductionCurrent reduction

For the fixed reference packing, the source makes optimality equivalent to | $N_*$ (A)|≥|A| for every finite nonoverlapping inserted collection A, where $N_*$ counts reference tetrahedra with interior conflict. A deficient collection gives an improving periodic replacement. Hall's theorem translates this condition into a matching but does not prove the geometric inequality.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve the finite-conflict inequality for every finite arbitrary packing, or exhibit one exact positive-count replacement.Task status · Ready to work on

Work mapped so far

Densest regular tetrahedron packing in numbers

1.1kretained lines of mathematical investigation1,129 in the current working snapshot

Argument development: 686 · 61%
Explored or eliminated routes: 36 · 3%
Computational analysis: 143 · 13%
Open obligations: 145 · 13%
Definitions and setup: 119 · 11%

9selected mapped statements2routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedSep 7, 2026

Source recordedSep 7, 2026

Recommended next task

Prove the finite-conflict inequality for every finite arbitrary packing, or exhibit one exact positive-count replacement.

Suggested move: Suggested next step: Choose a nontrivial replacement class, certify all interior conflicts and continuous pose coverage, and prove a collective Hall estimate with an explicit extension mechanism; alternatively verify every overlap condition for a deficient collection.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

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Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded

Narrowed routeUniversal lower bounds for raw centroid Voronoi cells or whole-face pairs

Route status · Narrowed route

Narrowed routeRepeating a favorable three-layer arrangement without checking the fourth layer

A specific first-to-fourth displacement lies strictly inside 2D; raw three-gap spans can equal 3±ε even at the reference. The fixed-lattice full-layer bound survives with corrected Y_i and applies to nonperiodic stacks; a changed layer class would need new hypotheses and coverage.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove the finite-conflict inequality for every finite arbitrary packing, or exhibit one exact positive-count replacement.Suggested move: Suggested next step: Choose a nontrivial replacement class, certify all interior conflicts and continuous pose coverage, and prove a collective Hall estimate with an explicit extension mechanism; alternatively verify every overlap condition for a deficient collection.

Ready to work on

Provide a global volume-transfer or cut estimate that handles unequal reference cells and arbitrarily long deficit regions.Suggested move: Suggested next step: Construct a finite-range antisymmetric correction or capacities and prove cancellation and every required region inequality without assuming layers or automatic whole-face pairing.

Ready to work on

Convert the unrestricted pair score into controlled support residuals or safe accounting without assuming pair hulls remain disjoint.Suggested move: Suggested next step: Prove a quantified near-dimer stability or signed-support decomposition theorem, and charge its overlap and compatibility errors without double counting.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026

Current statusOpen problem

The selected sources leave the unrestricted maximum density unresolved. Density 4000/4671 is achievable, while optimizing specific periodic families and proving a bound below one do not establish the exact global optimum.

[1][2][3]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2011
Peer reviewedGravel, Elser and Kallus proved a universal positive lower bound on uncovered space, hence a packing-density upper bound strictly below one. It is far from the construction density.[3]
2010
Peer reviewedChen, Engel and Glotzer constructed periodic double-dimer packings of density 4000/4671 and optimized a three-parameter family. That family restriction is essential.[1]
2010
Peer reviewedTorquato and Jiao gave exact constructions in a family of four-tetrahedron periodic packings and discussed the unresolved global optimization question.[2]

Mathematical neighborhood

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions2 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction1 of 91
lemma3 of 93
equivalence1 of 91
special case1 of 91
computational claim1 of 91
counterexample1 of 91

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.24 displayed rows · 2 routes included

retained route statementCan any packing of regular tetrahedra exceed density 4000/4671?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementFinite conflicts encode global optimalityintermediate
retained route statementStrict local optimality at each finite supercellintermediate
retained route statementSharp fixed-lattice full-layer theoremintermediate
retained route statementRotating-dimer certificate with full scope guardsintermediate
retained route statementNonnegative score for any disjoint pairintermediate
retained route statementExact obstructions to raw cell accountingintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported counterexample narrows one intermediate strategy; it does not challenge the open conjecture.challenges · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureUniversal lower bounds for raw centroid Voronoi cells or whole-face pairsreported failure
Useful failureRepeating a favorable three-layer arrangement without checking the fourth layerreported failure
Research targetProve the finite-conflict inequality for every finite arbitrary packing, or exhibit one exact positive-count replacement.open
Research targetProvide a global volume-transfer or cut estimate that handles unequal reference cells and arbitrarily long deficit regions.open
Research targetConvert the unrestricted pair score into controlled support residuals or safe accounting without assuming pair hulls remain disjoint.open
Narrowed routeUniversal lower bounds for raw centroid Voronoi cells or whole-face pairsThe exact cluster rules out a universal partition into singletons and whole-face dimers with a separate raw-cell lower bound v_* per tetrahedron. Compensated or grouped volume accounting must permit unequal reference cells and control all deficit regions.
Narrowed routeRepeating a favorable three-layer arrangement without checking the fourth layerA specific first-to-fourth displacement lies strictly inside 2D; raw three-gap spans can equal 3±ε even at the reference. The fixed-lattice full-layer bound survives with corrected Y_i and applies to nonperiodic stacks; a changed layer class would need new hypotheses and coverage.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve the finite-conflict inequality for every finite arbitrary packing, or exhibit one exact positive-count replacement.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

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Prepared starting pointProve the finite-conflict inequality for every finite arbitrary packing, or exhibit one exact positive-count replacement.

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Sources and references3 cited works · next context review by Dec 7, 2026

The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Dense crystalline dimer packings of regular tetrahedrapeer reviewed result · Elizabeth R. Chen, Michael Engel, Sharon C. Glotzer · Discrete & Computational Geometry 44, 253–280 · 2010 · ARXIV 1001.0586 · DOI 10.1007/s00454-010-9273-0 · accessed Sep 7, 2026
2
Exact Constructions of a Family of Dense Periodic Packings of Tetrahedrapeer reviewed result · S. Torquato, Y. Jiao · Physical Review E 81, 041310 · 2010 · ARXIV 1005.0011 · DOI 10.1103/PhysRevE.81.041310 · accessed Sep 7, 2026
3
Upper Bound on the Packing Density of Regular Tetrahedra and Octahedrapeer reviewed result · Simon Gravel, Veit Elser, Yoav Kallus · Discrete & Computational Geometry 46, 799–818; online 2010-10-14 · 2011 · ARXIV 1008.2830 · DOI 10.1007/s00454-010-9304-x · accessed Sep 7, 2026

Important qualifications

This is a scoped administrative literature search, not an exhaustive survey or an independent mathematical review.
Submitted packets are not external evidence. Their novel claims, corrections and proof obligations retain their separate source-reported posture.
No external code, proof development, data file or certificate was executed or independently reproduced during collection. Public links confer no acceptance or redistribution authority.
An empty formalization or recognition list means no suitable item was established in this search; it does not establish nonexistence. Unknown origin dates and original attributions are left unresolved.
The retained density is a proved construction value, not an independently audited claim to the latest world record. No complete contemporary packing-record census was performed.

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Densest regular tetrahedron packing

Exact mathematical statement

Problem at a glance

Where work on Densest regular tetrahedron packing stands

Densest regular tetrahedron packing in numbers

Prove the finite-conflict inequality for every finite arbitrary packing, or exhibit one exact positive-count replacement.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formal and computational footholds

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

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