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The packaging industry always wants to save money by using the least possible material in their boxes and packages. This example shows the smallest circle that can enclose 61 unit-sized circles. (The small circles are represented as marbles, for visual interest.) A shows the marbles arranged as four rings arranged evenly around a central marble. The red circle behind, the bounding circle, represents the area of the cylindrical box that would enclose them exactly, top and bottom. B shows the same marbles arranged in a similar way, but packed together as closely as possible, so that all marbles are in contact with other marbles. The result is a hexagon, but the bounding circle is identical to A. C is the mathematician's answer to the problem. Although the marbles are arranged very similarly to A, the red bounding circle has an area that is nearly 8% smaller than the other two. This is a 3D render.

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Free Images: "bestof:N-Exponentials Conjecture.png en An illustration of the logical implications between the circle of problems including the four exponentials conjecture and six"

Hadwiger conjecture.svg

Erdős–Faber–Lovász conjecture.svg

Hadwiger covering.svg

Keller-labeled Clebsch.svg

Double six.svg

N-Exponentials Conjecture.png

Comprehensive_Reference_on_Algebra_and_Equations_WDL2844.jpg

Musical Instruments and Musicians. Various Musical Instruments, Including a Koshnai WDL10771.png

Musical Instruments and Musicians. Various Musical Instruments, Including a Dutar, a Long-Necked Fretted Lute WDL10769.png

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Antiquities_of_Samarkand._Mausoleum_of_the_Emir_Timur_Kuragan_(Gur-Emir)._Inscription_along_the_Sides_and_inside_the_Main_Entry_Niche_WDL3725.png

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Strapping on Fire Wood.jpg

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US_Navy_050628-N-1464F-001_The_Italian_submarine_rescue_vehicle_SRV-300_is_launched_from_the_Italian_salvage_ship_Anteo.jpg

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US_Navy_050617-D-1825C-001_Chief_of_Naval_Operations,_Adm._Vern_Clark_speaks_at_the_completion_of_the_Naval_Postgraduate_School^rsquo,s_spring_quarter_in_Monterey,_Calif.jpg

US_Navy_050618-N-1464F-005_U.S._Navy_civilian_contractor_Nate_Kibbler,_top_left,_lowers_Dave_Diepenhorst_into_the_water_in_an_Atmospheric_Diving_System_(ADS_2000)_from_the_salvage_ship_USS_Grasp_(ARS_51).jpg

US_Navy_050620-N-1464F-026_Petty_Officer_3rd_Class_Richard_Russell_swims_out_to_the_United_Kingdom's_submarine_rescue_vehicle_LR-5_to_unhook_the_crane_as_it_was_lowered_off_the_Finnish_ship_Fennica.jpg

US_Navy_050620-N-1464F-040_U.S._Navy_Divers_in_a_Rigid_Hull_Inflatable_Boat_(RHIB)_assigned_to_Mobile_Diving_and_Salvage_Unit_Two_(MDSU2)_watch_the_United_Kingdom's_submarine_rescue_vehicle_LR-5.jpg

US_Navy_050624-N-8861F-001_U.S._Navy_Diver_Chief_Electrician's_Mate_Mike_Welsh_helps_Russian_Navy_Diver_Igor_Zhiazhivov_check_his_gear_before_the_dive_aboard_the_salvage_ship_USS_Grasp_(ARS_51).jpg

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US_Navy_070531-N-1928O-016_Seen_through_the_sights_of_a_World_War_II_era_40mm_anti-aircraft_gun,_guided_missile_frigate_USS_McInerney_(FFG_8)_transits_Boston_Harbor.jpg

ALFREDO GARCIA.jpg

Ketelaars ambassade bij vorst Udaipur Rijksmuseum NG-1987-7.jpeg

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Russian_Tu-144LL_SST_Flying_Laboratory_Landing_at_Zhukovsky_Air_Development_Center_DVIDS736990.jpg

Tu-144LL_SST_Flying_Laboratory_Lifts_off_Runway_on_a_High-Speed_Research_Flight_DVIDS692512.jpg

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New_floral_guide_(16862139802).jpg

Bazaar Types. Cane Vendor WDL10701.png

Places of Social Gathering. Tea House WDL10751.png

Musical Instruments and Musicians. Stringed Instruments and a Frame Drum WDL10770.png

Folk Festival during Ramadan. Hookah Smoker WDL10789.png

Folk Festival during Ramadan. Incense Vendor WDL10791.png

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Syr_Darya_Oblast._City_of_Turkestan._Khodzhalik,_a_Section_of_the_City_WDL10957.png

Syr_Darya_Oblast._City_of_Mankent._District_of_Cheshme_Bashi_WDL10934.png

Syr_Darya_Oblast._City_of_Tashkent._Russian_Section_of_City_WDL10946.png

Syr_Darya_Oblast._Aulie-Ata._Postal_Road_WDL10904.png

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Peddling._Barber_Shop_WDL10750.png

Places_of_Social_Gathering._Opium_Smoker_WDL10753.png

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