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    "title": "Dehn: equal volume does not guarantee a finite cut-and-reassemble solution",
    "summary": "Dehn established a necessary angle relation for finite polyhedral dissection into congruent pieces and proved that a regular tetrahedron cannot be reassembled into a prism, even when their volumes agree.\n\nTwo flat polygons of the same area can be cut into finitely many polygonal pieces and reassembled into one another. The analogous claim fails in three dimensions. Dehn used relations among edge lengths and dihedral angles to constrain any finite polyhedral dissection. A regular tetrahedron has a dihedral angle whose cosine is1/3 and which is not a rational multiple of pi; this obstructs the angle relations required for a rectangular box or prism. Thus equal volume alone cannot guarantee a finite reassembly. His1900 article constructs a normal form for a proposed dissection and proves necessary conditions; its language predates the later familiar tensor-product presentation of the Dehn invariant. Bricard had published an earlier approach in1896, which is retained as an antecedent rather than silently transferred to Dehn. The later classification that volume and Dehn invariant together suffice is a separate contribution of Sydler and subsequent simplifiers. The result concerns finite pieces that are polyhedra, not arbitrary nonmeasurable pieces, infinite limiting cuts or physical manufacture.",
    "boundary": "The obstruction concerns finite dissections of polyhedra into the stated class of pieces, preserving volume and the relevant angle/length invariant. Equal volume alone does not guarantee that such a dissection exists. This is not a prohibition of every infinite decomposition or arbitrary nonmeasurable construction. Hilbert presented the paper; that does not make him its coauthor.",
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        "description": "The obstruction concerns finite dissections of polyhedra into the stated class of pieces, preserving volume and the relevant angle/length invariant. Equal volume alone does not guarantee that such a dissection exists. This is not a prohibition of every infinite decomposition or arbitrary nonmeasurable construction. Hilbert presented the paper; that does not make him its coauthor.",
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      "statement": "# Dehn: equal volume does not guarantee a finite cut-and-reassemble solution\n\nDehn established a necessary angle relation for finite polyhedral dissection into congruent pieces and proved that a regular tetrahedron cannot be reassembled into a prism, even when their volumes agree.\n\nTwo flat polygons of the same area can be cut into finitely many polygonal pieces and reassembled into one another. The analogous claim fails in three dimensions. Dehn used relations among edge lengths and dihedral angles to constrain any finite polyhedral dissection. A regular tetrahedron has a dihedral angle whose cosine is1/3 and which is not a rational multiple of pi; this obstructs the angle relations required for a rectangular box or prism. Thus equal volume alone cannot guarantee a finite reassembly. His1900 article constructs a normal form for a proposed dissection and proves necessary conditions; its language predates the later familiar tensor-product presentation of the Dehn invariant. Bricard had published an earlier approach in1896, which is retained as an antecedent rather than silently transferred to Dehn. The later classification that volume and Dehn invariant together suffice is a separate contribution of Sydler and subsequent simplifiers. The result concerns finite pieces that are polyhedra, not arbitrary nonmeasurable pieces, infinite limiting cuts or physical manufacture.\n\n## Scientific scope\n\nThe obstruction concerns finite dissections of polyhedra into the stated class of pieces, preserving volume and the relevant angle/length invariant. Equal volume alone does not guarantee that such a dissection exists. This is not a prohibition of every infinite decomposition or arbitrary nonmeasurable construction. Hilbert presented the paper; that does not make him its coauthor.\n\n## People and systems\n\nM. Dehn\n\n\n\n## Original sources\n\n- [M.Dehn, Ueber raumgleiche Polyeder (1900) — original10page article](https://gdz.sub.uni-goettingen.de/download/pdf/PPN252457811_1900/LOG_0044.pdf)\n- [R.Bricard, Sur une question de geometrie relative aux polyedres (1896) — originalprecursor](https://www.numdam.org/item/NAM_1896_3_15__331_1.pdf)\n- [Pierre Cartier, Decomposition des polyedres (1986) — partiallaterreview](https://www.numdam.org/item/SB_1984-1985__27__261_0.pdf)\n",
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