![]() ![]() Fried (ME) AA 504: Fluid Mechanics Reviews the fundamentals with application to external and internal flows supersonic flow, 1D and Quasi-1D, steady and unsteady flow, oblique shocks and expansion waves, linearized flow, 2D flow, method of characteristics and transonic and hypersonic flow. Offered: Autumn (jointly with ME 503) Instructor: D. Offered: Autumn (even years) AA 503: Continuum Mechanics Reviews concepts of motion, stress, energy for a general continuum conservation of mass, momentum, and energy and the second law constitutive equations for linear/nonlinear elastic, viscous/inviscid fluids, and general materials and examples/solutions for solid/fluid materials. It is also possible for two sets of pieces to fit a rectangle of size 420, or for the set of 60 one-sided hexominoes (18 of which cover an even number of black squares) to fit a rectangle of size 360.Courses Offered by the Department of Aeronautics and Astronautics AA501 Physical Gasdynamics I Equilibrium kinetic theory chemical thermodynamics thermodynamic properties derived from quantum statistical mechanics reacting gas mixtures applications to real gas flows and gas dynamics. With checkerboard colouring, it has 106 white and 104 black squares (or vice versa), so parity does not prevent a packing, and a packing is indeed possible. For example, a 15 × 15 square with a 3 × 5 rectangle removed from the centre has 210 squares. However, there are other simple figures of 210 squares that can be packed with the hexominoes. However, any rectangle of 210 squares will have 105 black squares and 105 white squares, and therefore cannot be covered by the 35 hexominoes. Overall, an even number of black squares will be covered in any arrangement. If the hexominoes are placed on a checkerboard pattern, then 11 of the hexominoes will cover an even number of black squares (either 2 white and 4 black or vice versa) and the other 24 hexominoes will cover an odd number of black squares (3 white and 3 black). (Such an arrangement is possible with the 12 pentominoes which can be packed into any of the rectangles 3 × 20, 4 × 15, 5 × 12 and 6 × 10.) A simple way to demonstrate that such a packing of hexominoes is not possible is via a parity argument. Īlthough a complete set of 35 hexominoes has a total of 210 squares, it is not possible to pack them into a rectangle. This results in 20 × 8 + (6 + 2 + 5) × 4 + 2 × 2 = 216 fixed hexominoes.Įach of the 35 hexominoes satisfies the Conway criterion hence every hexomino is capable of tiling the plane. If rotations are also considered distinct, then the hexominoes from the first category count eightfold, the ones from the next three categories count fourfold, and the ones from the last category count twice. If reflections of a hexomino are considered distinct, as they are with one-sided hexominoes, then the first and fourth categories above would each double in size, resulting in an extra 25 hexominoes for a total of 60. It is the dihedral group of order 2, also known as the Klein four-group. The two purple hexominoes have two axes of mirror symmetry, both parallel to the gridlines (thus one horizontal axis and one vertical axis).Their symmetry group has two elements, the identity and the 180° rotation. ![]()
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