3Dmatrixen

3Dmatrixen is a term used to describe three-dimensional matrices, or three-dimensional arrays of numbers, commonly used in mathematics, computer science and engineering. They extend the concept of standard 2D matrices by adding a third index, so elements are identified as a i j k. In a practical setting, a 3Dmatrixen of size I × J × K contains I · J · K elements and can be viewed as a stack of I × J matrices along a third dimension, or as a sequence of K slices along the third axis. In abstract algebra this object is the same as a third-order tensor of order 3, though the term 3Dmatrixen is often used in applied contexts.

3Dmatrixen are stored in a linear memory array with a chosen order, such as row-major or column-major. Indexing uses three coordinates; operations may involve reducing across one or more axes, transposing axes, or reshaping. Element-wise operations are defined componentwise, while a general product requires a contraction over matching modes, or specialized products on slices (for example multiplying corresponding 2D matrices along a fixed axis).

Computational tools commonly treat 3Dmatrixen as tensors. In Python, NumPy uses ndarray objects with shape (I, J, K); PyTorch and TensorFlow extend this with automatic differentiation and GPU support. Typical workflows include slicing to access a fixed k-slice, computing along axes, broadcasting, and applying 3D convolutional operations in machine learning.

Applications include volumetric data representation (medical imaging, seismic data), computer graphics and 3D simulations, time-varying data sets, and neural networks that operate on 3D inputs. A simple example is a 2 × 2 × 2 array of numbers used to illustrate indexing, where an element is retrieved as a[i, j, k].

See also tensors, multi-dimensional arrays, and tensor contraction.