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The synthetic dimension opens new horizons in quantum physics and topological photonics by enabling new dimensions for field and particle manipulations. The most appealing property of the photonic synthetic dimension is its ability to emulate high-dimensional optical behavior in a unitary physical system. Here we show that the photonic synthetic dimension can transform technical problems in photonic systems between dimensionalities, providing unexpected solutions to technical problems that are otherwise challenging. Specifically, we propose and experimentally demonstrate a photonic Galton board (PGB) in the temporal synthetic dimension, in which the temporal high-speed challenge is converted into a spatial fiber-optic length matching problem, leading to the experimental generation of tera-sample-per-second arbitrary waveforms. Limited by the speed of the measurement equipment, waveforms with sampling rates of up to 341.53 GSa/s are recorded. Our proposed PGB operating in the temporal synthetic dimension breaks the speed limit in a physical system, bringing arbitrary waveform generation into the terahertz regime. The concept of dimension conversion offers possible solutions to various physical dimension-related problems, such as super-resolution imaging, high-resolution spectroscopy, time measurement, etc.