Despite being one of the earliest neural network layers, the Multilayer Perceptron (MLP) is arguably one of the least understood parts of the transformer architecture due to its dense computation and lack of easy visualization. This paper seeks to understand the MLP layers in dense LLM models by hypothesizing that these layers secretly approximately perform a sparse computation -- namely, that they can be well approximated by sparsely-activating Mixture of Experts (MoE) layers. Our hypothesis is based on a novel theoretical connection between MoE models and Sparse Autoencoder (SAE) structure in activation space. We empirically validate the hypothesis on pretrained LLMs, and demonstrate that the activation distribution matters -- these results do not hold for Gaussian data, but rather rely crucially on structure in the distribution of neural network activations. Our results shine light on a general principle at play in MLP layers inside LLMs, and give an explanation for the effectiveness of modern MoE-based transformers. Additionally, our experimental explorations suggest new directions for more efficient MoE architecture design based on low-rank routers.