Zonse polyhedra: zinthu zodabwitsa ndi madera

Masamu ndi wokongola chifukwa chakuti, mosiyana ajebura, omwe si nthawi zonse bwino chifukwa ndi zimene mukuganiza, amapereka chinthu zithunzi. dziko lino zodabwitsa za matupi osiyanasiyana kukometsera polyhedra zonse.

General mudziwe polyhedra zonse

Malinga ndi ambiri, polyhedrons zonse, kapena monga iwo amatchedwa Plato zolimba, adzalandira wapadera katundu. Ndi awa zinthu chikugwirizana malingaliro angapo sayansi. Pamene muyamba kuphunzira deta zojambula za thupi, inu mukuzindikira kuti pafupifupi sindikudziwa kanthu za mfundo monga polyhedra zonse. Ulaliki wa zinthu izi mu sukulu si nthawi zonse chidwi, ambiri sakudziwa nkomwe kukumbukira zimene iwo anali kutchedwa. Pokumbukira anthu ambiri ndi chabe kyubu. Palibe Masamu thupi alibe ungwiro monga polyhedrons zonse. Maina onse a matupi awa zojambula linachokera ku Greece wakale. Akuimira chiwerengero cha nkhope: ndi tetrahedron - anayi amaganiza, hexahedron - Allen, octahedron - octagon, dodecahedron - dodecahedral, icosahedron - icosahedral. Onsewa thupi zojambula ukutchulidwa kwambiri mu lingaliro Plato chilengedwe. Anai a iwo ophatikizidwa zinthu kapena mabungwe: ndi tetrahedron - moto, icosahedron - madzi kyubu - lapansi, octahedron - mpweya. Dodecahedron zili zonse. Iye ankaona waukulu, monga chizindikiro cha chilengedwe.

The generalization ndi maganizo polyhedron ndi

Polyhedron ndi deta amalire wa polygons ngati kuti:

• aliyense wa m'mbali mwa aliyense wa polygons ndi pa nthawi yomweyo mbali imodzi yokha polygon wina kumbali imodzi;
• kwa aliyense wa polygons mukhoza kuyenda kwa wina ndi kudutsa pafupi thereto polygons.

Polygons amapanga polyhedron kuimira nkhope yake ndi mbali yawo - nthiti. polyhedra mfundo ndi mfundo za polygons. Ngati akuti polygon kumvetsa lathyathyathya polylines chatsekedwa nkubwera tanthauzo lina la polyhedron a. Ngati kumene akuti imeneyi inaperekedwa gawo la ndege kuti njanjira ndi mizere wosweka, adzakhululukidwa anamvetsa pamwamba wopangidwa zidutswa polygonal. Otukukira kunja polyhedron amatchedwa thupi atagona pa mbali imodzi ya ndege, pafupi nkhope yake.

Wina tanthauzo la polyhedron ndi zanyengo

Polyhedron otchedwa pamwamba ongokhala polygons, zomwe zimasokoneza thupi zojambula. Ali:

• sanali otukukira kunja;
• otukukira kunja (zabwino ndi zoipa).

Zonse polyhedron - ndi otukukira kunja polyhedron ndi yayikulu chosokonekera. Zochitika za polyhedra zonse:

• Tetrahedron: 6 nthiti 4 nkhope 5 mfundo;
• hexahedron (kyubu) 12, 6, 8;
• dodecahedron 30, 12, 20;
• octahedron 12, 8, 6;
• icosahedron 30, 20, 12.

Lingaliro Lovomerezeka Euler a

Iwo amakhazikitsa ubale pakati pa chiwerengero cha m'mbali, mfundo ndi nkhope ndi topologically lofanana dera wina. Kuwonjezera chiwerengero cha mfundo ndi nkhope (B + D) ndi osiyana polyhedra zonse ndipo anawayerekezera ndi chiwerengero cha nthiti, ndi zotheka ulamuliro wina: Uwerenge chiwerengero cha nkhope ofanana kuchuluka kwa mfundo ndi m'mbali (P) chinawonjezeka ndi 2. Nkotheka amapeza chilinganizo losavuta:

• B + D = P + 2.

chilinganizo zimenezi zili zoyenera kwa polyhedra onse otukukira kunja.

matanthauzo zoyambirira

Lingaliro la polyhedron zonse n'zosatheka kufotokoza m'chiganizo chimodzi. Ndi zambiri ofunika ndi buku. Thupi amadziwidwa monga, m'pofunika kuti akukumana angapo matanthauzo. Motero, thupi zojambula adzakhala polyhedron zonse pamene zinthu izi zimapezeka:

• ndi otukukira kunja;
• nambala yomweyo nthiti lokhazikika pa aliyense wa mfundo zake;
• mbali zake zonse za - polygons zonse, wofanana ndi mzake;
• kumathandiza kupeza ngodya zabwino dihedral ofanana.

Katundu wa polyhedra zonse

Pali 5 osiyanasiyana polyhedra zonse:

1. Kyubu (hexahedron) - izo ndi tayala pamwamba ngodya ndi 90 °. Iwo ali ngodya 3 amaganiza. Kuchuluka nkhope kumathandiza kupeza ngodya zabwino pa pamwamba 270 °.
2. Tetrahedron - lathyathyathya pamwamba ngodya ya - 60 °. Iwo ali ngodya 3 amaganiza. Kuchuluka nkhope kumathandiza kupeza ngodya zabwino pa pamwamba - 180 °.
3. Octahedron - lathyathyathya pamwamba ngodya ya - 60 °. Iwo ali ngodya zinayi amaganiza. Kuchuluka nkhope kumathandiza kupeza ngodya zabwino pa pamwamba - 240 °.
4. Dodecahedron - lathyathyathya pamwamba ngodya ya 108 °. Iwo ali ngodya 3 amaganiza. Kuchuluka nkhope kumathandiza kupeza ngodya zabwino pa pamwamba - 324 °.
5. Icosahedron - iwo ali lathyathyathya pamwamba ngodya ya - 60 °. Iwo ali ngodya asanu amaganiza. Kuchuluka nkhope kumathandiza kupeza ngodya zabwino pachimake pa ° 300.

M'dera la polyhedra zonse

M'dera padziko matupi geometrical (S) kuchita masamu ngati zonse polygon m'dera kuchulukitsa ndi chiwerengero cha mbali (G):

• S = (a: 2) X 2G ctg π / p.

Buku la polyhedron zonse

Khalidwe limeneli ndi kuchita masamu ndi kuchulukitsa buku la piramidi zonse amene m'munsi ndi polygon zonse, chiwerengero cha nkhope, ndipo msinkhu wake ndi utali wozungulira alembedwa a dera (R):

• V = 1: 3rS.

Buku la polyhedra zonse

Ngati wina zojambula olimba, polyhedra zonse ndi mabuku osiyanasiyana. M'munsimu muli mitunduyi ndi zomwe iwo akhoza kuwerengera:

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

Zochitika za polyhedra zonse

Hexahedron ndi octahedron ndi wapawiri matupi zojambula. M'mawu ena, iwo akhoza kutuluka mzake kukachitika kuti centroid wa munthu walandiridwa monga pamwamba pa ena, ndi mosemphanitsa. Komanso ali wapawiri icosahedron ndi dodecahedron. Yekha tetrahedron ndi wapawiri. Malinga ndi njira Euclid mukhonza kuwapeza ndi dodecahedron hexahedron ndi kumanga "madenga" pa nkhope za kyubu lapansi. The mfundo za tetrahedron ndi aliyense mfundo 4 kyubu, osati awiriawiri moyandikana pamodzi m'mphepete. Kuyambira hexahedron (kyubu) mukhonza kuwapeza, ndi zina polyhedra zonse. Ngakhale kuti polygons zonse pali osawerengeka, zonse polyhedra, pali 5 okha.

The radii wa polygons zonse

Ndi aliyense wa matupi awa zojambula ndi zikugwirizana concentric muzochitika 3:

• anafotokoza kudutsa mfundo ya;
• yomangira lililonse la nkhope yake pakati pa izo;
• apakatikati za m'mbali zonse pakati.

The utali wozungulira dera anafotokozedwa ndi chilinganizo zotsatirazi kuchita masamu:

• R = ndi: 2 × tg π / g × tg θ: 2.

The utali wozungulira dera lolembedwa ndi kuchita masamu motere:

• R = ndi: 2 × ctg π / tsa × tg θ: 2,

kumene θ - dihedral ngodya zapakati mbali pafupi.

The utali wozungulira apakatikati a dera akhoza kuchita masamu pogwiritsa ntchito mfundo zotsatirazi:

• ρ = ndi Ko π / tsa: 2 tchimo π / h,

kumene H = ukulu wa 4.6, 6,10, kapena 10. chiŵerengero cha radii wa alembedwa anafotokoza ndi symmetrically ndi ulemu kwa tsamba ndi Q. Iwo masamu motere:

• R / r Mukhoza = tg π / tsa × tg π / Q.

The chosokonekera wa polyhedra

The chosokonekera wa polyhedra zonse ndi chidwi chachikulu kwa matupi awa zojambula. Iwo anazindikira monga kayendedwe ka thupi danga, imene masamba zochuluka mofana ndi mfundo, nkhope ndi m'mbali. M'mawu ena, mchikakamizo chosokonekera masinthidwe m'mphepete, vertex, kapena nkhope akuzisunga pamalo ake pachiyambi, kapena chimachititsanso kunyumba malo a nthiti wina, mfundo zina kapena nkhope.

Zochitika chosokonekera wa polyhedra zonse sizachilendo kwa mitundu yonse ya zolimba zojambula. Apa imachitika pa kusintha dzina, imene masamba mfundo ina iliyonse mu malo pachiyambi. Choncho, pamene inu kutembenuzira prism polygonal mukhoza kupeza ena symmetries. Aliyense wa iwo akhoza ankayimiridwa ngati mankhwala a chinyezimiritso. Chosokonekera, amene ali chopangidwa ndi chiwerengero cha mawunikidwe, wotchedwa mwachindunji. Ngati ndi mankhwala a wosamvetseka angapo mawunikidwe, ndiye amatchedwa maganizo. Choncho, ankasinthana onse padziko mzere akuyimira molunjika chosokonekera. Aliyense chinyezimiro polyhedron - ndi osiyanitsidwa chosokonekera.

Kumvetsetsa zinthu zodabwitsa za polyhedra zonse, inu mukhoza kutenga chitsanzo cha tetrahedron lapansi. mzere aliyense amene adzadutsa imodzi mwa mfundo ndi pakati pa mawonekedwe zojambula, zidzachitika, ndi pakati pa mzinda wa m'mphepete zosiyana kwa izo. Aliyense wa ankasinthana 120 ndi 240 ° padziko mzere wa mochulukitsa tetrahedral chosokonekera. Popeza 4 mfundo ndi nkhope, ife okwana asanu ndi atatu symmetries mwachindunji. Aliyense wa mizere adapyola pakati m'mbali ndi likulu la thupi, umadutsa pakatikati pa m'mphepete zosiyana. kasinthasintha aliyense wa 180 °, wotchedwa theka-kutembenuka ndi chosokonekera molunjika. Popeza tetrahedron ali magulu atatu a nthiti, inu kupeza zifukwa zitatu chosokonekera. Zochokera kumwamba, tinganene kuti chiwerengero cha chosokonekera mwachindunji, ndipo kuphatikizapo kusintha dzina, lidzakhala kwa khumi ndi ziwiri. Other mwachindunji chosokonekera tetrahedron kulibe, koma 12 osiyanitsidwa chosokonekera. Choncho, 24 yodziwika tetrahedron symmetries. Chifukwa ndi losavuta kumva, tingalimbitse chitsanzo cha tetrahedron zonse chidapangidwa kuchokera ku katoni ndipo onetsetsani kuti ndi thupi zojambula alidi 24 chosokonekera.

Dodecahedron ndi icosahedron - pafupi ndi malo thupi. Icosahedron ali ndi chiwerengero chachikulu cha nkhope, mbali dihedral ndipo koposa zonse akhoza mwamphamvu amamatira ku dera alembedwa. Dodecahedron ali otsika okhota chilema waukulu olimba ngodya pa vertex lapansi. Iwo akhoza kudzapeza lembani dera circumscribed.

chindodo polyhedra

Zonse polyhedra jambulani, tonse ogwirizana ana, ndi zambiri mfundo. Ngati pali ya polygons, mbali iliyonse imene imafotokozedwa ndi mbali imodzi ya polyhedron, ndi chizindikiritso cha zipani ayenera kutsatira zinthu ziwiri:

• aliyense polygon, inu mukhoza kupita kwa polygon ndi chizindikiritso cha mbali;
• kuuzindikira mbali ayenera kutalika.

Ndi ya polygons kuti pali zinthu izi, ndipo akutchedwa polyhedron jambulani. Aliyense wa matupi awa ali angapo a iwo. Mwachitsanzo, kyubu amene pali zidutswa 11.