Ka pololei, polyhedra: hehee wale symmetry a me ka wahi

Anahonua mea maikai no ka mea, e like hōʻailona helu, i mea e mau maopopo ke kumu a me ka mea e manaʻo, haawi i ka pilikia makapō mea. He kupanaha honua o mau kino Ua wehi i ka mau polyhedra.

General 'ike ma regular polyhedra

E like me ia he nui loa, regular polyhedrons, a me ka mea i kapaʻia Platonic puaʻa hiwa, e komo kū hoʻokahi waiwai. Me kēia mau mea e pili ana kekahi mau akeakamai hypotheses. Ka wā e hoʻomaka e hoʻopaʻa haʻawina i ka geometric ikepili o ke kino, e ike i aneane mai i ike i kekahi mea e pili ana ia i ka manaʻo, e like me ka mea mau polyhedra. ʻO ka hōʻike no kēia mau mea i loko o ke kula mea,ʻaʻole e mau ana hana hoihoi, no laila, he nui mai i hiki hoomanao i ka mea i kapaʻia. I ka paʻa loa kanaka o ia mea like he kupa. None o ke kino anahonua, aole ia e lilo i ka mea maikaʻi a me ka pololei, polyhedrons. ka inoa o kēia mau geometric kino a pau originated mai kahiko Greece. Ka mea, ho i ka helu o nā maka: ka tetrahedron - eha-kapakahi, hexahedron - Allen, octahedron - octagon, dodecahedron - dodecahedral, icosahedron - icosahedral. A pau o kēia mau geometric kino ninoaaeyao nui i wahi i loko o Plato ka hapai ana o ka ke ao holoʻokoʻa. Eha o ia i hookomo oihana mua a me nāʻoihana: ka tetrahedron - i ke ahi, i ka icosahedron - wai kupa - honua, octahedron - ea. Dodecahedron hookomo nā mea a pau. He Ua noʻonoʻo i ka papa kuhikuhi, e like me ka hōʻailona o ka ke ao holoʻokoʻa.

Ke generalization o ka manaʻo o kekahi polyhedron

Polyhedron mea he finite ohi o polygons ia ia:

• kela mea keia mea o na aoao elua o kekahi o na polygons mea i ka hookahi manawa wale kekahiʻaoʻao o kekahi iieeaii ma kaʻaoʻao hookahi;
• mai kela a me keia o na polygons oe e hele i ka mea'ē aʻe e hele e pili ana polygons.

Polygons aha ka polyhedron ho i kona maka a me kā lākou mauʻaoʻao - iwi aoao. polyhedra vertices i na vertices o polygons. Inā ke kau iieeaii maopopo i lalo paʻa polylines, alaila, e hele mai i kekahi ka wehewehena o ka polyhedron. Ma ka hihia ai ma keia makahiki i manaoia i kekahi hapa o ka pelane i ua kau palena ia ana e wawahi ai i ka laina, ka mea e e hoomaopopo ili oia hoi o polygonal apana. Convex polyhedron ua kapaia ke kino, e moe ana ma kekahiʻaoʻao o ka pelane, e pili ana i kona mau maka.

Kekahi ka wehewehena o ka polyhedron a me kona oihana mua

Polyhedron kapa ili oia hoi o polygons, a hoʻohaiki ana i ka geometric kino. Ka mea, e:

• ole-convex;
• convex (pono a me ka hewa).

Ka pololei, polyhedron - mea he convex polyhedron me maximal symmetry. Oihana mua o regular polyhedra:

• Tetrahedron: 6 iwi aoao 4 maka 5 vertices;
• hexahedron (kupa) 12, 6, 8;
• dodecahedron 30, 12, 20;
• octahedron 12, 8, 6;
• icosahedron 30, 20, 12.

Euler ka theorem

Ua hoʻokau i ka pilina ma waena o ka helu o nā kihi, vertices a me nā maka i topologically like me ka poepoe. Ohui i ka helu ana o vertices a me nā maka (B + D) i okoa regular polyhedra a hoʻohālikelikeʻia ia me ka helu o nā iwi, he mea hiki, e hoʻonoho i kekahi rula: i ka huina o ka helu o nā maka e like me ka helu ana o vertices a me nā kihi a (P), a mahuahua nui e 2. He mea hiki ke loaa i ka mea haʻilula:

• B + D = P + 2.

Kēia haʻilula mea i pololei ia no nā mea a pau convex polyhedra.

walaʻauʻana wehewehena i

Ka manaʻo o kekahi mau polyhedron mea hiki ole e wehewehe mai i kekahi olelo. He oi kuaiia a me ka leo. A kino e eʻike me ia, ka mea i pono ia ia me he helu o ka wehewehena i. Penei, he geometric kino, e ia i ka papa polyhedron ka wā mau hana hea halawai.

• ia mea convex;
• ka hookahi helu o nā iwi aoao converges ma kela a me keia o kona vertices;
• a pau facets o kona - regular polygons, like i kekahi i kekahi;
• All dihedral pana pua e like.

Waiwai o regular polyhedra

Aia i 5ʻano o mau polyhedra:

1. Kupa (hexahedron) - ka mea, ua i lalo apex huina o 90 °. Ua loaʻa he 3-kapakahi huina. Huina maka pana pua ma ka apex o ka 270 °.
2. Tetrahedron - i lalo apex huina o - 60 °. Ua loaʻa he 3-kapakahi huina. Huina maka pana pua ma ka apex - 180 °.
3. Octahedron - i lalo apex huina o - 60 °. Ua loaʻa he eha-kapakahi huina. Huina maka pana pua ma ka apex - 240 °.
4. Dodecahedron - he i lalo apex huina o 108 °. Ua loaʻa he 3-kapakahi huina. Huina maka pana pua ma ka apex - 324 °.
5. Icosahedron - ka mea, ua i lalo apex huina o - 60 °. Ua loaʻa he elima-kapakahi huina. Huina maka pana pua ma ka apex o ka 300 °.

Ka wahi o ka pololei, polyhedra

Kaʻilikai kahi o ka geometrical kino (S) he pōpilikia me he mau iieeaii āpana hoonuiia e ka helu ana o facets (G):

• S = (he: 2) x ka 2G ctg π / p.

Ka buke o ka mau polyhedron

Kēia waiwai he pōpilikia ma ka hoonui i ka buke o ka papa pyramid kona kumu o ka regular iieeaii, ka helu o nā maka, a me kona kiekie o ka kākauʻia ke kahahńnai o ka poepoe (R):

• V = 1: 3rS.

Puke a regular polyhedra

Like kekahi'ē aʻe geometric paa, regular polyhedra i kekahi mau puke. Aia ma lalo nō nā papakuhikuhi ma i ka mea hiki ke ho omaulia:

• Tetrahedron: α m 3√2: 12;
• octahedron: α m 3√2: 3;
• icosahedron; α m 3;
• hexahedron (kupa): α m 5 m 3 m (3 + √5): 12;
• dodecahedron: α m 3 (15 + 7√5): 4.

Oihana mua o regular polyhedra

Hexahedron a me ka octahedron i pālua geometric kino. Ma na olelo e ae, ka mea e hele mai ana kekahi i kekahi i loko o ka hanana i ka centroid o kekahi e ia e like me ka luna o ka mea'ē aʻe, a me ka hope versa. No hoi i pālua icosahedron a me ka dodecahedron. Oia wale tetrahedron Ua pālua. E like me ke ano o Euclid hiki ke loaa mai ka dodecahedron hexahedron ma ka kūkulu "puhi" ma ka maka o ke kupa. Nā vertices o ka tetrahedron i kekahi 4 vertices o ke kupa,ʻaʻole e pili hui ma ka lihi. Mai hexahedron (kupa) hiki ke loaa, a me nā maʻamau polyhedra. I loko nō o ka mea e mau polygons , ua pau i ka heluia, regular polyhedra, loaʻa nō 5 wale.

Ke Radii o mau polygons

Me kela mea keia mea o kēia mau geometric kino e pili ana concentric spheres 3:

• ho'ākāka 'ia ma maalo ae la oia ma na vertices;
• kākauʻia e pili ana i kela mea keia mea o kona maka ma ka waenakonu o ka ia;
• Media e pili ana i na kihi o ka waena.

Ke kahahńnai o ka poepoe i hoakakaia ma ke kēia haʻilula he pōpilikia:

• R = i: 2 He m tg π / g. He m tg θ: 2.

he pōpilikia ke kahahńnai o ka kākauʻia poepoe like penei:

• R = i: 2 He m ctg π / pa m tg θ: 2,

kahi θ - dihedral huina i mea ma waena o e pili facets.

Ka Media ke kahahńnai o ka poepoe hiki ke ho omaulia hoʻohana 'ana i nā kēia haʻilula:

• ρ = he cos π / pa: 2 hewa π / h,

kahi? h = ka nui o 4,6, 6,10, a me 10. ka lākiō o ka Radii o ka kākauʻia i hoakakaia a symmetrically me ka mahalo i ka pa a me ka Q. Ua he pōpilikia like penei:

• R / R = tg π / pa m tg π / Q.

Ke symmetry o polyhedra

Ke symmetry o ka regular polyhedra mea o iniiaiie hoihoi i kēia mau geometric kino. Ma ka hoʻomaopopo aku me ka holo ana o ke kino ma ka makahiki, i na lau o ka hookahi helu o vertices, maka a me nā lua penei. Ma nā hua'ōlelo, ma lalo o ka aoao o ka symmetry loli 'maka, vertex, a me ka maka hoopaa mai ai i kona palapala kulana, a e neʻe aku i ka hale oihana o kekahi iwiʻaoʻao, ka mea'ē aʻe vertices paha i lalo ke alo.

Oihana mua o symmetry o ka regular polyhedra mea, he pono ole i na mea a pau ano o geometric puaʻa hiwa. Eia ka mea, ua mālama 'ia ma luna o ka wahine paha, loli, i na lau i kekahi o nā kumu i loko o ka palapala kulana. No laila, i ka wa e huli ai ka polygonal prism ke kiʻi i kekahi mau symmetries. Kekahi o ia mau mea hiki e na poe like me ka huina hoonui o ka noʻonoʻo. Symmetry, i mea no ka huina hoonui o ka i ka helu o ka pepa hōʻuluʻulu, kapa pololei. Inā ka mea, o ka huina hoonui o ka ōlaʻi helu o nā manaʻo, a laila ka mea, ua kapaia manaʻo. Ke'ī mai, a pau nā 'ana e haʻalele puni ka laina ho i pololei symmetry. I kekahi mau manaʻo polyhedron - o ka inverse symmetry.

E pono maopopo i ka symmetry hehee wale o ka regular polyhedra, oe i ke kumu hoohalike o ke tetrahedron hiki ke lawe. Kekahi laina e, e hele ana oukou ma kekahi o na vertices a me ka waena konu o ka geometric kinona, e lawe wahi, a ma ka waenakonu o ka maka e ku pono ana ia ia. Kēlā me kēia no nā 'ana e haʻalele 120 a me 240 ° a puni o ka laina e pili ana i ka Plural tetrahedral symmetry. No ka mea, 4 vertices a me nā maka, ua loaa i ka huina o kaʻewalu pololei symmetries. Kekahi o na laina maalo ae la oia ma ka waena o nā kihi a me ka waena konu o ke kino, ka mea hele ma ka waena o ka mea ku pono lihi. Kekahi kuapo o 180 °, i kapaʻia ka hapa-huli puni pololei ka symmetry. Mai ka tetrahedron i ekolu hui o ka iwiʻaoʻao ona, e kiʻi laina ekolu o ka symmetry. Ma muli o ka mea ma luna, ua hiki ke manao nei makou, o ka huina helu o ka ala symmetry, a me ka wahine paha, loli, e e mai i ka umikumamalua. Other kauoha symmetry tetrahedron hakuʻole, akā, ka mea, he 12 inverse symmetry. Nolaila, 24 wale wehewehe aku tetrahedron symmetries. No ka mōakaaka, ua hiki ke kūkulu i kekahi kumu hoʻohālike o ka papa tetrahedron i ka pepa, a e hōʻoia i ka mea o ka geometric kino maoli he 24 wale symmetry.

Dodecahedron a icosahedron - hea ka mea me ka kino kahi. Icosahedron i ka nui helu o nā maka, ka dihedral huina a me ka hapanui o nā mea a pau e paa i hoʻopili i ke kākauʻia poepoe. Dodecahedron i ka lalo angular kina nui paa huina ma ka vertex. Ua hiki ke hoʻonui loa, e hoʻopiha i ka circumscribed poepoe.

kaʻimiʻana polyhedra

Regular polyhedra scan NineManga.com, a mākou i nā mea a pau apo la a pau i loko o kamaliʻi, i ka hailona o ka manaÿo. Inā he mea kekahi lākou o nā polygons, ua hōʻike 'kēlā me kēiaʻaoʻao o ka a me ka wale kekahiʻaoʻao o ka polyhedron, e hoʻokō i ka' ike o nā 'aoʻao a me nā ana:

• o kela a me keia iieeaii, e hiki ke hele i ke iieeaii me ka 'ike o kaʻaoʻao;
• i moakākaʻaoʻao e i ka ia loa.

He He lākou o nā polygons e halawai mau rula, a ua kapaʻia he polyhedron scan NineManga.com. Kēlā me kēia no kēia mau kino loaʻa paha kekahi mau no ia. No ka laʻana, he kupa no a loaʻa he 11 apana.