The paper aims to apply the trigintaduonion spaces to explore the physical properties of four interactions simultaneously, including the electromagnetic fields, gravitational fields, weak nuclear fields, and strong nuclear fields. J. C. Maxwell first applied the algebra of quaternions to study the physical properties of electromagnetic fields. It inspired some subsequent scholars to introduce the quaternions, octonions, sedenions, and trigintaduonions to research the electromagnetic fields, gravitational fields, weak nuclear fields, strong nuclear fields, quantum mechanics, gauge fields, and curved spaces and so forth. The algebra of trigintaduonions is able to discuss the physical quantities of four interactions, including the field potential, field strong, field source, linear momentum, angular momentum, torque, and force. In the field theories described with the algebra of trigintaduonions, the weak nuclear field is composed of three types of fundamental fields. These three fundamental fields, related to weak nuclear fields, can describe the physical properties of weak nuclear fields collectively. This is consistent with the conclusion of the electroweak theory. Meanwhile the strong nuclear field consists of three types of fundamental fields. These three fundamental fields relevant to strong nuclear fields may investigate the physical properties of strong nuclear fields mutually. It is coincident with the deduction of quark theory. According to the properties of trigintaduonions, one can deduce the Yang-Mills equation related to the gauge fields. It means that the electromagnetic field occupies a quaternion space. The gravitational field owns one different quaternion space. The weak nuclear fields occupy three mutually independent quaternion spaces. The properties of weak nuclear fields are different from those of electromagnetic fields or gravitational fields. According to the multiplicative closure, the strong nuclear fields also own three quaternion spaces independent of each other. These explorations further deepen the understanding of the physical properties of weak and strong nuclear fields.

The paper aims to explore the exponential forms of octonion angular momenta in the electromagnetic and gravitational fields, researching the influencing factors and physical properties of octonion wavefunctions. J. C. Maxwell first utilized the quaternions and vectorial terminology to describe the electromagnetic theory. Nowadays, the scholars apply the quaternions and octonions to study the electromagnetic fields, gravitational fields, and quantum mechanics and so forth. The application of octonions is able to describe the physical quantities of electromagnetic fields and gravitational fields, including the octonion field strength, field source, linear momentum, angular momentum, torque, and force and others. According to the characteristics of octonions, the octonion physical quantities can be rewritten into the exponential forms. In particular, either the angular momentum or electromagnetic moment may be dominant under certain circumstances, in the octonion spaces. The product of the octonion angular momentum and Planck's constant can constitute a nondimensionalized octonion exponential form. As a result, the octonion wavefunctions can be obtained from the exponential forms of octonion angular momenta. When the direction of multidimensional unit vector in the octonion wavefunction cannot be determined, the imaginary unit can be used to substitute the multidimensional unit vector. As a compensation measure, it is necessary to replace one single octonion wavefunction, relevant to a multidimensional unit vector, with several wavefunctions related to the imaginary units. The dimension number of unit vector may be interrelated to the color number of color charges in the quantum chromodynamics.

The paper focuses on applying the octonions to explore the influence of the external torque on the angular momentum of fluid elements, revealing the interconnection of the external torque and the vortices of vortex streets. J. C. Maxwell was the first to introduce the quaternions to study the physical properties of electromagnetic fields. The contemporary scholars utilize the quaternions and octonions to investigate the electromagnetic theory, gravitational theory, quantum mechanics, special relativity, general relativity and curved spaces and so forth. The paper adopts the octonions to describe the electromagnetic and gravitational theories, including the octonionic field potential, field strength, linear momentum, angular momentum, torque and force and so on. In case the octonion force is equal to zero, it is able to deduce eight independent equations, including the fluid continuity equation, current continuity equation, and force equilibrium equation and so forth. Especially, one of the eight independent equations will uncover the interrelation of the external torque and angular momentums of fluid elements. One of its inferences is that the direction, magnitude and frequency of the external torque must impact the direction and curl of the angular momentum of fluid elements, altering the frequencies of Karman vortex streets within the fluids. It means that the external torque is interrelated with the velocity circulation, by means of the liquid viscosity. The external torque is able to exert an influence on the direction of downwash flows, improving the lift and drag characteristics generated by the fluids.