A fast algorithm is established to transform points of the unit sphere into fundamental region symmetrically. With the resulting algorithm, a flexible form of invariant mappings is achieved to generate aesthetic patterns with symmetries of the regular polyhedra. This method avoids the order restriction of symmetry groups, which can be similarly extended to treat regular polytopes in n-dimensional space for n>=4.

Folk residence in the cold region of Northeast China is an important part of China's traditional folk residence, with significant geographical representation. In this paper, we take the residential compounds of villages and towns in the cold region of Northeast China as the research object, summarize four typical layout forms of residential compounds of villages and towns in the cold region of Northeast China, simulate the wind environment of compounds of different layout forms in winter by using the computer numerical simulation software Fluent, and then analyze and evaluate the wind environment of compounds by combining with the evaluation standard of wind speed ratio and considering the area of wind shadows, number of vortices and other factors.

The stator winding insulation system is the most critical and weak part of the EMU's (electric multiple unit) traction motor. The effective diagnosis for stator ITSC (inter-turn short-circuit) faults can prevent the fault from expanding into phase-to-phase or ground short-circuits. The TCU(traction control unit) controls the traction inverter to output SPWM (sine pulse width modulation) excitation voltage when the traction motor is stationary. Three ITSC fault diagnostic conditions are based on different IGBTs control logic. The Goertzel algorithm is used to calculate the fundamental current amplitude difference Δi and phase angle difference Δθ of equivalent parallel windings under the three diagnostic conditions. The six parameters under the three diagnostic conditions are used as features to establish an ITSC fault diagnostic model based on random forest. The proposed method was validated using a simulation experimental platform for ITSC fault diagnosis of EMU traction motors. The experimental results indicate that the current amplitude features Δi and phase angle features Δθ change obviously with the increase of ITSC fault extent if the ITSC fault occurs at the equivalent parallel windings. The accuracy of the ITSC fault diagnosis model based on the random forest for ITSC fault detection and location both in train and test samples are 100%.