Have you ever wondered why financial markets seem so unpredictable and counterintuitive? Why do patterns that look obvious on a 2D price chart evaporate when you add more variables? Or why building a robust trading model feels like chasing shadows—the more data you incorporate, the harder it gets?

The answer might lie in the quirky world of higher-dimensional mathematics. I recently came across a fascinating post by Alexander Kruel outlining “dimensional anomalies”—ways in which geometric and algebraic patterns we take for granted in 2D or 3D worlds break down or transform dramatically in higher dimensions. These aren’t just abstract curiosities; they mirror the challenges of modeling financial markets, which are inherently high-dimensional systems. Markets aren’t flat charts; they’re vast spaces shaped by countless interacting factors: prices, volumes, economic indicators, sentiment, geopolitical events, and more. Each new variable adds a “dimension,” and as we’ll see, high dimensions bring surprises that make market behavior feel alien and non-intuitive.

In this post, I’ll connect Kruel’s mathematical insights to trading model development, highlighting why our low-dimensional intuitions fail and how embracing dimensionality can lead to better (or at least more humble) strategies. Let’s dive in.

1. The Explosion of Possibilities… Then Collapse

Kruel starts with regular polytopes: In 2D, there are infinitely many regular polygons. Jump to 3D, and you’re down to just 5 Platonic solids. In 4D, it’s 6, but by 5D and higher, only 3 families remain.

This “peak and decline” echoes the feature selection dilemma in trading models. In low-dimensional setups—like a simple model using just price and volume (think 2D)—you have endless ways to slice the data: moving averages, RSI, Bollinger Bands, and so on. It feels flexible and intuitive. But as you add dimensions (e.g., incorporating macroeconomic data, options implied volatility, or social media sentiment), the “regular” patterns—reliable, symmetric relationships—become scarce. Suddenly, your model overfits to noise because the space is too vast for your dataset to cover evenly.

Trading takeaway: High dimensionality prunes viable strategies. What works in a simplistic 2D backtest (e.g., a crossover strategy) often crumbles when you layer in real-world variables. Focus on “surviving families” like robust, low-parameter models (e.g., trend-following in multiple assets) that generalize across dimensions.

2. Tilings and the Search for Perfect Market Coverage

In 2D, there are 3 regular tilings (triangles, squares, hexagons). In 3D, just one (cubic). Back up to 3 in 4D, then down to one in higher dimensions.

Markets are like trying to “tile” the data space with patterns or rules. In low dimensions, you can cover the landscape neatly—think grid-based strategies or uniform risk allocation across a few assets. But in high-dimensional markets (where “tiles” are decision boundaries in machine learning models), perfect coverage is rare. Most strategies leave gaps or overlaps, leading to inefficiencies or blowups.

Non-intuitive market observation: Why do diversified portfolios sometimes fail spectacularly? In high dimensions, the “honeycomb” of correlations isn’t uniform; shocks propagate unevenly, like irregular tilings. This explains black swan events—your model assumes a cubic 3D structure, but markets are more like a sparse 5D lattice.

3. Knots: Entanglements That Vanish in Higher Dimensions

Kruel notes that knots don’t exist in 2D, abound in 3D, but untangle easily in 4D and above. (Though higher knots shift to surfaces.)

In trading, “knots” are entangled risks or correlations—like how interest rates, inflation, and stock prices twist together in a 3D macro model. Our brains, wired for 3D intuition, see these as permanent snarls (e.g., “bonds and stocks always move inversely”). But add dimensions—like global supply chains or crypto influences—and these knots can “slip free.” What seemed like a locked-in arbitrage opportunity unravels.

Why markets feel non-intuitive: Traders get burned assuming low-dimensional permanence. In reality, high-dim markets allow for unlinking: A knotty trade setup (e.g., yield curve inversion predicting recession) might dissolve with new variables like AI-driven productivity booms.

4. The Vanishing Volume: Where’s the Meat in High-Dimensional Data?

The volume of a unit ball increases initially but then shrinks toward zero as dimensions go to infinity. Most “mass” hugs the surface.

This is the curse of dimensionality in action for trading. In low-dim models (e.g., pairs trading two stocks), data fills the space densely. But in high-dim (e.g., a neural net with 100 features), volume explodes, and your finite dataset becomes a thin shell—sparse and unrepresentative. Points cluster near the edges, making anomalies (outliers) the norm.

Trading implication: Overfitting is inevitable because “most of the space” is empty. Market observations like “this pattern held 90% of the time” are misleading; in high dims, rare events dominate the tails, explaining fat-tailed returns and why Value at Risk models underestimate crashes.

5. Kissing Numbers and Sphere Packings: Crowding in Market Spaces

How many spheres touch a central one? 6 in 2D, 12 in 3D, jumps to 24 in 4D, with huge leaps in 8D and 24D.

In markets, “spheres” are assets or strategies; the “kissing number” is how many can correlate without overlapping destructively. In low dims, like a portfolio of 3-5 stocks, you can pack them tightly (diversify efficiently). But in high dims—multifactor models—the optimal packing is unknown or jumps unpredictably, thanks to symmetric “lattices” like factor models (e.g., Fama-French).

Densest packings? Proven in 2D/3D, but only recently in 8D/24D. Markets mirror this: Simple pairs trading (2D hexagonal) works okay, but high-frequency trading in multidimensional order books requires exotic “lattices” like those in quant funds.

Non-intuitive twist: In high dims, you can “pack” more strategies (e.g., 196,560 in 24D analogy), but most are inefficient. This explains why quant blowups happen—crowding into suboptimal packings leads to liquidity evaporating like spheres jamming.

6. Waves, Algebras, and Cross Products: Lost Properties in High Dims

Huygens’ principle for sharp waves only in odd dims (3,5+). Division algebras in 1,2,4,8D. Cross products only in 3 and 7D.

These highlight lost symmetries: Commutativity, associativity, orthogonality fade away. In trading, low-dim models assume nice properties—like linear regressions where variables “commute” (order doesn’t matter). But high-dim markets lose this: Non-linear interactions (e.g., options gamma vs. spot price) aren’t associative, leading to path-dependent outcomes.

Why non-intuitive? Our 3D brains expect cross-product-like intuitions (e.g., “momentum crosses value”), but in 10D factor spaces, they don’t exist. Markets “wave” with tails in even dims, explaining lingering volatility after shocks.

7. The Curse and Random Walks: Why Markets Wander Off

The curse: Space explodes, distances concentrate, vectors orthogonal. Random walks recurrent in 1-2D, transient in 3D+.

Markets are random walks in high dims—prices drift transiently, rarely returning to means. Distances concentrate: All assets seem “equally far” in correlation space, making diversification illusory. Orthogonality: Factors that seem independent are nearly so, but tiny alignments cause cascades.

Core non-intuitiveness: Low-dim thinking assumes recurrence (mean reversion), but high-dim markets are transient—trends persist or explode. This is why “buy the dip” works until it doesn’t, and sparsity curses your dataset: 10^10 samples needed for 10 features with 10 bins each.

Wrapping Up: Embracing the Dimensional Weirdness

These mathematical anomalies reveal why markets defy intuition: We’re applying 2D/3D mental models to 10D+ realities. Trading isn’t about finding “truths” that hold universally; it’s navigating phase changes where rules flip. To build better models:

• Dimensionality reduction: Use PCA or autoencoders to collapse to survivable dims.  
• Embrace sparsity: Regularization techniques (LASSO) counter the curse.  
• Test for anomalies: Simulate high-dim behaviors to avoid low-dim biases.  
• Humility in intuition: Recognize that like knots untying or volumes vanishing, market patterns are dimension-dependent.

Financial markets are a high-dimensional jungle—beautiful, but brutal if you ignore the anomalies. Next time a trade goes sideways, remember: It might just be the dimension talking.

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