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プラトンの固体とアーキメデスの固体の密度の高い包装
1Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA. torquato@princeton.edu
Nature
|August 14, 2009
まとめ
研究者は,密度の高い多面体包装を調査し,プラトンの固体の新しい最も密度の高い構成を発見しました. この研究は,ケプラーと同様の対称的多面体のための最適な格子パッケージを提案しています.
科学分野:
- 離散幾何学による離散幾何学.
- 数学の理論は,数学の理論である.
- マテリアルサイエンス 材料科学
- 計算物理学の物理
背景:
- 密度の高い粒子のパッケージは,物質や材料の様々な状態をモデル化しています.
- 研究は主に球形粒子に焦点を当てており,多面体パッケージに関する知識は限られている.
- パッキング対称性を理解することは,離散幾何学や数論において極めて重要です.
研究 の 目的:
- 非タイルプラトン型固体 (四面体,八面体,十二面体,二面体) に対して知られている最も密度の高い包装を決定する.
- 多面体パッキング密度と格子構造との関係を調査する.
- プラトニックとアーキメデスの対称的な固体の最も密度の高い包装に関する推測を提案する.
主な方法:
- オプティマイゼーション問題として,多面体パッキング生成を策定した.
- 周期的な境界条件を持つ適応的基本細胞 (適応的縮小細胞) を採用した.
- シミュレーションのために多様な多粒子の初期構成を使用した.
主要な成果:
- 四面体 (0.782...),八面体 (0.947...),十二面体 (0.904...),二面体 (0.836...) のパックに記録密度を達成しました.
- 八面体,十二面体,イコサヘドラの最も密度の高い包装は,それらの既知の最適な格子包装に対応することを発見しました.
- 最も密度の高い四面体包装は,Bravaisではないことが確認されました.
結論:
- 非タイル性プラトニック固体の最も密度の高い包装が特定されています.
- 一つの推測が提案されています:対称的なプラトンとアーキメデスの固体の最も密度の高い包装は,それらの最も密度の高い格子包装です.
- この発見は,ケプラーの球の推測を,多面体固体のより広いクラスに拡張します.
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