A Mathematician’s Guided Tour Through Higher Dimensions
Author: David S. Richeson
Publication date: September 19, 2021
Original publication:Quanta Magazine, September 13, 2021
Source:WIRED / Quanta Magazine
The article explores how the seemingly obvious idea of dimension became a surprisingly deep mathematical concept. We intuitively think of a point as $0$-dimensional, a line as $1$-dimensional, a plane as $2$-dimensional, and ordinary space as $3$-dimensional. Higher dimensions can be approached through analogy: just as a three-dimensional sphere passing through a two-dimensional world would appear as a changing sequence of circles, a four-dimensional object passing through our space could be observed only through changing three-dimensional cross-sections. The four-dimensional analogue of a cube, the tesseract, can similarly be understood by extending the progression point $\to$ line $\to$ square $\to$ cube into an additional independent direction.
But the intuitive definition of dimension began to break down in the nineteenth century. Georg Cantor showed that the points of a line segment and those of a square can be placed in one-to-one correspondence, meaning that cardinality alone cannot distinguish their dimensions. Giuseppe Peano then constructed a continuous curve capable of filling an entire square. These results forced mathematicians to formulate a rigorous concept of dimension. In 1912, L. E. J. Brouwer established the invariance of dimension, showing that Euclidean spaces of different dimensions are genuinely topologically distinct. In modern notation, spaces such as $\mathbb{R}^2$ and $\mathbb{R}^3$ cannot be topologically equivalent.
The story becomes even stranger with Felix Hausdorff's concept of dimension, which allows dimensions that are not integers. For a self-similar object scaled by a factor $k$, one can think of its size as scaling roughly like $k^d$, where $d$ is its dimension. The Koch curve, for example, consists of four smaller copies of itself, each scaled by $1/3$, so its dimension satisfies
$3^d=4$,
giving
$d=\frac{\log 4}{\log 3}=\log_3 4\approx1.26$.
Such fractional dimensions became fundamental to Mandelbrot's theory of fractals. The article closes by connecting higher-dimensional mathematics with relativity, string theory, sphere packing, statistics, and machine learning, where the rapid growth of complexity with dimension produces the famous “curse of dimensionality.”
Key takeaways
ARTICLE / ARTICLE [PDF]
Author: David S. Richeson
Publication date: September 19, 2021
Original publication:Quanta Magazine, September 13, 2021
Source:WIRED / Quanta Magazine
The article explores how the seemingly obvious idea of dimension became a surprisingly deep mathematical concept. We intuitively think of a point as $0$-dimensional, a line as $1$-dimensional, a plane as $2$-dimensional, and ordinary space as $3$-dimensional. Higher dimensions can be approached through analogy: just as a three-dimensional sphere passing through a two-dimensional world would appear as a changing sequence of circles, a four-dimensional object passing through our space could be observed only through changing three-dimensional cross-sections. The four-dimensional analogue of a cube, the tesseract, can similarly be understood by extending the progression point $\to$ line $\to$ square $\to$ cube into an additional independent direction.
But the intuitive definition of dimension began to break down in the nineteenth century. Georg Cantor showed that the points of a line segment and those of a square can be placed in one-to-one correspondence, meaning that cardinality alone cannot distinguish their dimensions. Giuseppe Peano then constructed a continuous curve capable of filling an entire square. These results forced mathematicians to formulate a rigorous concept of dimension. In 1912, L. E. J. Brouwer established the invariance of dimension, showing that Euclidean spaces of different dimensions are genuinely topologically distinct. In modern notation, spaces such as $\mathbb{R}^2$ and $\mathbb{R}^3$ cannot be topologically equivalent.
The story becomes even stranger with Felix Hausdorff's concept of dimension, which allows dimensions that are not integers. For a self-similar object scaled by a factor $k$, one can think of its size as scaling roughly like $k^d$, where $d$ is its dimension. The Koch curve, for example, consists of four smaller copies of itself, each scaled by $1/3$, so its dimension satisfies
$3^d=4$,
giving
$d=\frac{\log 4}{\log 3}=\log_3 4\approx1.26$.
Such fractional dimensions became fundamental to Mandelbrot's theory of fractals. The article closes by connecting higher-dimensional mathematics with relativity, string theory, sphere packing, statistics, and machine learning, where the rapid growth of complexity with dimension produces the famous “curse of dimensionality.”
Key takeaways
- Dimension is much more subtle than simply counting coordinates. Cantor and Peano showed why naive definitions fail.
- Brouwer's invariance-of-dimension theorem supplied a rigorous mathematical distinction between spaces such as $\mathbb{R}^2$ and $\mathbb{R}^3$.
- Hausdorff dimension can be fractional, allowing mathematicians to quantify the complexity of fractals such as the Koch curve.
- Higher-dimensional mathematics is not merely theoretical: it appears in relativity, data science, statistics, machine learning, fractals, and modern geometry.
ARTICLE / ARTICLE [PDF]
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