YANKAN ARCHIMEDES SUNA AIKI NE KAWAI A KAN ƘWALLO

Takardar ta fara da wani wasan kwakwalwa: shin za a iya rufe faifai mai faɗin raka’a 100 da ɗamarori marasa iyaka guda 99, kowanne faɗinsa raka’a 1? Amsar, in ji Mijia Lai na Makarantar Kimiyyar Lissafi a Jami’ar Shanghai Jiao Tong, “tana mayar da mu fiye da shekaru dubu biyu baya zuwa ɗaya daga cikin lokutan Eureka na Archimedes”.

Ɗamara da girman filinta ba ya kula da inda take

Ka’idar Archimedes. A kan ƙwallo mai radius 1, yankin da ke tsakanin shimfiɗu biyu masu layi ɗaya waɗanda dukansu suka yanka ƙwallon, tazararsu h, yana da girman fili 2πh.

Abin mamaki shi ne abin da dabarar ta bari: inda ɗamarar take. Kusa da ma’auni (equator) ko kusa da sanda (pole), tazarar h kaɗai ke da muhimmanci.

Alamar da takardar ta bayar don wasan kwakwalwar ita ce a ɗaga ɗamarorin daga faifai mai faɗi zuwa kan ƙwallo, inda suka zama ɗamarori a kanta. Bin wannan alama tare da dabarar yana ba da amsar. Ƙwallo mai faɗin raka’a 100 tana da radius 50 da girman fili 4π × 50² = 10,000π. Kowace ɗamara mai faɗin raka’a 1 tana rufe 2π × 50 × 1 = 100π. Guda tasa’in da tara suna rufe 9,900π a mafi yawa — bai isa ba. Don haka a’a, ɗamarorin ba za su iya rufe faifan ba.

Tambayar da aka juya

Shin wannan siffa tana keɓe ƙwallo kaɗai? Ga fuskoki convex — siffofi marasa lotsawa — kuma idan ana buƙatar siffar a kowace tazara, an daɗe da sanin amsar; takardar ta danganta ta da Blaschke. Masanin lissafi Mohammad Ghomi ya yi tambaya mafi tsauri a dandalin MathOverflow a Oktoba 2017: idan girman filin da ke tsakanin shimfiɗu biyu masu layi ɗaya iri ɗaya ne don tazara guda ɗaya tak, a duk lokacin da dukan shimfiɗun suka yanka fuskar convex, dole ne wannan fuskar ta zama ƙwallo?

Lai ya tabbatar da cewa dole ne, kuma ya ƙara gaba.

Ka’ida A. Ɗauki fuska santsi, rufaffiya, haɗaɗɗiya a sararin samaniya. A ce akwai tazara h, ƙasa da mafi ƙanƙantar faɗin fuskar, wadda kowane yanki mai faɗin h — a kowace alkibla da kowane wuri inda dukan shimfiɗun suka yanka fuskar — yana da girman fili 2πh. To fuskar ƙwallo ce mai radius 1.

Ba a ɗauka cewa convex ce ba: hakan wani ɓangare ne na sakamakon.

Matakai biyu, ratsawa ɗaya ta raƙuman ruwa

Na farko, babu lotsawa. Sharaɗin yankan yana da sakamako na siffa: inda fuskar za ta lanƙwasa ciki, nesa da lulluɓinta na waje, yana tilasta wani adadi da ake kira matsakaicin lanƙwasa (mean curvature) ya zama sifili ko ƙasa da sifili. Lai ya nuna cewa wannan yana kai ga saɓani, ta amfani da wani tsohon kayan aiki na irin waɗannan lissafai, ƙa’idar mafi girma ta Hopf (Hopf maximum principle). Idan babu lotsawa, dole fuskar ta zama convex, kuma a zahiri ta lanƙwasa waje a kowane wuri.

Sannan, ƙwallo kaɗai. Ga fuskar convex, sharaɗin yankan yana sa yadda girman filin ke rarrabuwa bisa tsawo ya maimaita kansa kowace h. Ɗaukar matsakaicin wannan tsari mai maimaituwa a dukan alkibloli yana samar da maganin lissafin raƙumi (wave equation) da ke maimaituwa a lokaci. Ƙimarsa ta farko ta zama tana da alaƙa da ƙarfin Newton (Newtonian potential) — irin wanda nauyi ko cajin lantarki ke samarwa — wanda fuskar za ta samar idan an shafa ta daidai-wa-daida. Wani sakamako na keɓancewa ga irin waɗannan raƙuman ruwa masu maimaituwa, da aka tabbatar a ƙari (appendix), sannan yana tilasta wannan ƙarfi ya zama tsayayye a ciki da wajen fuskar. Wata sananniyar ka’idar daidaito, ta Reichel, ta ce ƙwallo mai ƙarfi kaɗai ce za ta iya samar da irin wannan ƙarfi. A ƙarshe dabarar Archimedes ta saita radius ɗinta zuwa 1.

Wani sashe na daban, mai sauƙi, ya yi bayani kan yanayi na musamman inda faɗin fuskar a wata alkibla cikakken adadi ne na tazarar h.

Taimako daga AI

A cikin godiyarsa, marubucin ya gode wa tsarin AI ChatGPT 6 Astra “saboda taimako kan dabarun hujja”, ya ƙara da cewa an samo hujjar mai sauƙi ta sashe na ƙarshe kafin wannan taimakon, kuma ya ɗauki alhakin dubawa da rubuta hujjar. Sakamakon preprint ne kuma ba a riga an yi masa bitar takwarori ba.

Daga wasan kwakwalwa game da ɗamarori a kan faifai zuwa raƙuman ruwa masu maimaituwa da fuskoki masu caji, hujjar ta ƙare inda Archimedes ya fara: ɗamara mai faɗin h, da girman fili 2πh.