| gptkbp:instanceOf | gptkb:mathematical_concept gptkb:fractal
 gptkb:energy_spectrum
 
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                                | gptkbp:appearsIn | two-dimensional electron systems 
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                                | gptkbp:dependsOn | gptkb:lattice_structure magnetic flux
 
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                                | gptkbp:describedBy | gptkb:Douglas_Hofstadter 
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                                | gptkbp:exhibits | self-similarity fractal structure
 
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                                | gptkbp:firstDescribed | 1976 
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                                | gptkbp:hasApplication | gptkb:cold_atom_systems gptkb:graphene_superlattices
 quantum materials
 optical lattices
 
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                                | gptkbp:hasFeature | band structure energy gaps
 magnetic field dependence
 quasiperiodicity
 topological invariants
 
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                                | gptkbp:hasModel | gptkb:Harper's_equation 
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                                | gptkbp:namedAfter | gptkb:Douglas_Hofstadter 
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                                | gptkbp:publishedIn | gptkb:Physical_Review_B 
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                                | gptkbp:relatedTo | gptkb:Harper's_equation gptkb:tight-binding_model
 gptkb:quantum_Hall_effect
 
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                                | gptkbp:studiedIn | condensed matter physics mathematical physics
 
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                                | gptkbp:symmetry | self-similarity fractal symmetry
 
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                                | gptkbp:visualizes | fractal pattern 
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                                | gptkbp:bfsParent | gptkb:Douglas_Hofstadter 
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                                | gptkbp:bfsLayer | 5 
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                                | https://www.w3.org/2000/01/rdf-schema#label | Hofstadter butterfly 
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