Simetrijske grupe končnih vzorcev

2021 ◽

Author(s):

Matej Mencinger

Keyword(s):

Končni vzorci so najprej definirani intuitivno, kasneje pa še eksaktno (matematično). Simetrije končnih vzorcev so definirane s pomočjo izometrij ravnine. Obravnavani so štirje osnovni razredi izometrij ravnine: zrcaljenje, rotacija, translacija ter drsno zrcaljenje. V klasifikacijskem izreku je dokazano, da vsaka izometrija spada v enega od osnovnih štirih razredov. Pri obravnavi vektorjev in matrik se omejimo na ravnino in trirazsežni vektorski prostor. Dokazano je, da izometrijam ravnine s fiksno točko pripadajo natanko ortogonalne matrike. V poglavju o grupah so obravnavani pojmi: (pod)grupa, red grupe, izomorfizem grup ter generatorji grupe. Glavni rezultat je klasifikacija simetrijskih grup končnih vzorcev v ciklične ali diedrske, kar danes imenujemo Leonardov izrek. Podano je tudi nekaj informacij o zgodovinski dobi, v kateri je deloval Leonardo da Vinci in nekaterih povezavah med njegovim delom in matematiko. Učbenik je namenjen študentom arhitekture in vsebuje številne primere, rešene naloge ter obsežno slikovno gradivo.

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Replication of Internal Biological Surfaces

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100094802 ◽

1972 ◽

Vol 30 ◽

pp. 308-309

Author(s):

J. A. Nowell ◽

J. Pangborn ◽

W. S. Tyler

Keyword(s):

Da Vinci ◽

Leonardo Da Vinci ◽

Rat Lung ◽

Tubular Structures ◽

Cerebral Ventricles ◽

Internal Surfaces ◽

Biological Surfaces ◽

Dog Kidney ◽

Lung And Kidney ◽

Scanning Electron

Leonardo da Vinci in the 16th century, used injection replica techniques to study internal surfaces of the cerebral ventricles. Developments in replicating media have made it possible for modern morphologists to examine injection replicas of lung and kidney with the scanning electron microscope (SEM). Deeply concave surfaces and interrelationships to tubular structures are difficult to examine with the SEM. Injection replicas convert concavities to convexities and tubes to rods, overcoming these difficulties.Batson's plastic was injected into the renal artery of a horse kidney. Latex was injected into the pulmonary artery and cementex in the trachea of a cat. Following polymerization the tissues were removed by digestion in concentrated HCl. Slices of dog kidney were aldehyde fixed by immersion. Rat lung was aldehyde fixed by perfusion via the trachea at 30 cm H2O. Pieces of tissue 10 x 10 x 2 mm were critical point dried using CO2. Selected areas of replicas and tissues were coated with silver and gold and examined with the SEM.

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