Armine Grigoryan
Posts
Post has attachment
Post has shared content
Post has shared content
Post has shared content
Post has attachment
Post has shared content
4 cubes can be glued trivially face-to-face in R^3 forming a ring, their centers being the vertices of a square. We have seen, however, that only in R^4 can we have a ring of 3 cubes (glued face-to-face), their centers being the vertices of an equil. trig; what was more, we saw that it was possible (in R^4) to have 4 cubes (always glued f.-to-f. through squares) whose centers are the vertices of a reg. tetrahedron.
What happens with reg. tetrahedra? As a matter of fact it is impossible to glue in R^3 reg. tetrahedra face-to-face (through equil. trigs) forming a closed ring; the best we can do is gluing 5 tetrah. face-to-face in R^3 in the said manner without forming a ring [in fact the dihedral angle between the faces of a tetrah. being 70.5 deg, we can never complete 360 deg around an edge of a reg. tetrah.
What happens in R^4? Is it possible to form a ring of tetrahedra that closes in R^4? and how many tetrah. (always glued in the said manner) are needed to form such a closed ring?
The animation herewith shows that such a ring is easy to obtain in R^4 & that it is formed of 4 reg. tetrah. glued face-to-face through equil. trigs.
the hypersolid I present is a convex one in R^4 & is formed by a 4-ring of reg. tetrahedra (in light blue); in the development (or net) I use, there appear to be a gap in the ring, but that is due to the manner the reg tetrah. were unwrapped from R^4 to R^3( such that they all fit in R^3, which is impossible without showing a gap) [the 2 equil. trigs are labled both 1-2-4 & are actually glued in R^4 between themselves (in a ring every tetrah. must be glued simultaneously to 2 other ones (the one which preceeds & the one which follows): this gluing uses 2 faces of each tetrahedra, so that leaves 2 faces free in each tetrah. Those 2 free faces are glued to 2 trirectangular tetrahedra so that the hypersolid is composed of 4 reg. tetrahedra (the 4-ring) + 8 trt's all with a common vertex labeled 7. To put things in another way: the hypersolid of which you see the development in R^3 is the cumulation of 4 tetrahedric 4-pyramids whose bases are reg. tetrahedra & lateral faces are trt tetrah.
The reg tetrah. that form the ring are: 1-2-3-4, 1-2-3-5, 1-2-4-6 & 1-2-5-6 ; the 1st is covered in red & maroon, the 2nd in pink & purple, the 3rd in blue & light green & the 4th in yellow & dark green.
For those who might enjoy having the coordinates of the 7 vertices in the fig., they are: (2,0,0,0),(0,2,0,0),(0,0,2,0),(0,0,0,2),(0,0,0,-2),(0,0,-2,0) & (0,0,0,0) the last the origin of coordinates and the common vertex to the 8 trt's.
Please be well, the whole pack!... (lol)
Post has attachment
Post has shared content
Сегодня день Интернета в России.
Приглашаю поиграть вместе!
Начало игры-викторины
в 22 часа по Московскому времени!
Играем в режиме онлайн!
Тема:
Что мы знаем об ИНТЕРНЕТ
1. Откройте http://join.quizizz.com в своем браузере
2. Введите 6-значный код игры, 925321
и нажмите "Продолжить"
3. Введите свое имя и нажмите
"Вступить в игру!"
4.
Вы получите свой аватар, а затем увидите кнопку "Начать". Нажмите её, чтобы приступить к игре!
Всего вы увидите 19 вопросов, где один из 3 или 4 вариантов верен.
Оценивается верный и быстрый ответ.
На обдумывание 30 секунд.
Таким образом вся игра займет 10-15 минут!