NeuroHex: A Brain-Inspired Hex Coordinate System to Enable Highly Computationally-Efficient World Models for Continuous Online-Adaptive Learning

Quinn Jacobson; Joe Luo; Jingfei Xu; Shanmuga Venkatachalam; Kevin Wang; Dingchao Rong; John Paul Shen

cs.AIFeb 27, 2026

NeuroHex: A Brain-Inspired Hex Coordinate System to Enable Highly Computationally-Efficient World Models for Continuous Online-Adaptive Learning

Authors: Quinn Jacobson, Joe Luo, Jingfei Xu, Shanmuga Venkatachalam, Kevin Wang, Dingchao Rong, John Paul Shen

Organizations: NeuroAI Computer Architecture Lab (NCAL), Carnegie Mellon University

Abstract

NeuroHex is a brain-inspired hexagonal coordinate system designed to support highly efficient world models and reference frames for online adaptive AI systems. Inspired by the hexadirectional firing structure of grid cells in the human brain, NeuroHex adopts a cubic isometric hexagonal coordinate formulation that provides full 60° rotational symmetry and low-cost translation, rotation and distance computation. We develop a mathematical framework that incorporates ring indexing, quantized angular encoding, and a hierarchical library of foundational, simple, and complex geometric shape primitives. These constructs allow low-overhead point-in-shape tests and spatial matching operations that are expensive in Cartesian coordinate systems. To support realistic settings, we also develop a novel tool (OSM2Hex) that can process OpenStreetMap (OSM) data sets and convert them into the NeuroHex coordinate system. The OSM2Hex spatial abstraction processing pipeline can achieve a reduction of 90-99% in geometric complexity while maintaining the relevant spatial structure map for navigation. Our initial results, based on actual city and neighborhood scale data sets, demonstrate that NeuroHex offers a highly efficient substrate for building dynamic world models to enable adaptive spatial reasoning in autonomous energy-efficient AI systems with continuous online-adaptive learning (COAL) capability.

Figures & tables

Fig. 1: Hexagonal Coordinate Systems: H3Geo (left) with positive coordinates (i,j,k) [ 23 ] ; our NeuroHex (right) with positive and negative coordinates (q,r,s).

Fig. 2: High-Level working model of the brain with both Egocentric and Allocentric perspectives involving Grid Cells, adaptive World Models, and hierarchy of Reference Frames.

Fig. 3: Hierarchical Reference Frames used for building maps of physical world based on diverse sensory modalities.

Fig. 4: Foundational Components in the NeuroHex coordinate system: Ordered Wedge, Ring, and Point.

Fig. 5: Wedge Partitioning Based On Sign-based Wedge Indexing

Fig. 6: Distance Computation (left) and Angular Direction Quantization (right) in the NeuroHex coordinate system.

Fig. 7: Foundational Shapes (left) with single reference point, and Simple Shapes involving two reference points (right).

Fig. 8: Examples of Translation, Rotation, and Scaling of shapes in the NeuroHex coordinate system.

Fig. 9: Variable Resolutions of Hexagonal Tiles in NeuroHex.

Operation	NeuroHex Coordinate System	Two-Dimensional Computational Geometry
Distance ${}^{*\textit{(a)}}$ \raisebox{-0.9pt}{1}⃝	$\max\!\left(\|q-q^{\prime}\|,\ \|r-r^{\prime}\|,\ \|s-s^{\prime}\|\right)$	$\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}}$
Radial Distance	$\max\!\left(\|q\|,\ \|r\|,\ \|s\|\right)$	$\sqrt{x^{2}+y^{2}}$
Denormalization	$(R_{ws}\cdot R_{i}^{\textit{(b)}})\gg B^{\textit{(c)}}$	$\phi_{ref1}\cdot\phi_{ref2}\gg B$
Polar Angle $\mathbf{(\phi_{point}^{\textit{(d)}})}$ \raisebox{-0.9pt}{2}⃝	$R_{ws}+W_{i}^{\textit{(e)}}R_{i}$	$\operatorname{atan2}(x,y)$
Sign Decoding ∗ \raisebox{-0.9pt}{3}⃝	$(q,r,s)=\bigl(s_{q}\|q\|,\ s_{r}\|r\|,\ s_{s}\|s\|\bigr)$ $\mathcal{S}(\text{Wedge})=(s_{q},s_{r},s_{s})$ $s_{q},s_{r},s_{s}\in\{-1,0,+1\}$	$(x,y)=\bigl(s_{x}\|x\|,\ s_{y}\|y\|\bigr)$ $\mathcal{S}(\text{Quadrant})=(s_{x},s_{y})$ $s_{x},s_{y}\in\{-1,0,+1\}$
Orientation Predicate	Obtain $\phi_{1}$ and $\phi_{2}$ via \raisebox{-0.9pt}{2}⃝ and determine whether $\phi_{1}<\phi_{2}$	$\det\!\begin{pmatrix}x_{1}-x_{2}&y_{1}-y_{2}\\ x-x_{2}&y-y_{2}\end{pmatrix}>0$

TABLE I: Comparison of computation complexities between NeuroHex system and Cartesian computational geometry.

Kernel	FF	LUT
NeuroHex Distance	50	240
Cartesian Distance	104	885
NeuroHex Polar Angle	53	352
Cartesian Polar Angle	821	1419
NeuroHex Rotate	142	660
Cartesian Rotate	1057	4797

TABLE II: Post-synthesis Resource Utilization of NeuroHex and Cartesian Compute Kernels from Vitis suggests that the NeuroHex implementation is most competent for primitives dealing with angle displacement, averaging 4 to 10 times less FPGA fabric usage. On the other hand, Cartesian kernels are more cost-efficient for operations that exploit edge functions, with a slight lead on PIT test and 2x resource efficiency for orientation predicate.

Name	Slice LUTs	Slice Registers	F7 Muxes	F8 Muxes	Slice	LUT as Logic	DSP	Power (W)
NeuroHex Distance	28	50	0	0	20	28	0	¡0.001
Cartesian Distance	191	91	0	0	96	191	0	0.004
NeuroHex Polar Angle	54	53	0	0	0	0	0	0.001
Cartesian Polar Angle	351	475	0	0	0	0	0	0.003
NeuroHex Rotation	143	84	0	0	0	0	0	0.001
Cartesian Rotation	830	465	0	0	0	0	0	0.001

TABLE III: Post-implementation Resource Utilization and Power Report from Vivado displays similar trends to those shown in Table 2, taking into account the added usage of DSP. The power report indicates that NeuroHex outperforms Cartesian with a steady 3x-4x less power consumption in most cases but falls behind on orientation predicate.

Kernel	Type	Add		Sub		Mul		Div		Cmp		Sqrt
Kernel	Type	NH	C	NH	C	NH	C	NH	C	NH	C	NH	C
op_distance	combined	0	2	6	2	0	4	0	0	8	0	0	1
	16-bit	0	0	6	2	0	2	0	0	8	0	0	1
	32-bit	0	1	0	0	0	2	0	0	0	0	0	0
	10-bit fixed	0	1	0	0	0	0	0	0	0	0	0	0
op_polar_angle	combined	1	6	2	6	1	4	0	4	7	14	0	0

TABLE IV: Arithmetic Operation Counts for NeuroHex (NH) and Cartesian (C) Compute Kernels With 4-bit Quantization. The algorithmic structure of geometric operations is consistent with the observed utilization of resources. NeuroHex excels at converting geometric problems into integer selection, wedge localization, and bounded comparison.

Fig. 10: Complete end-to-end data flow of the OSM2Hex system. Raw OpenStreetMap data is acquired via the Overpass API, parsed, classified, geometrically simplified, triangulated, and merged into NeuroHex-compatible shape primitives. At the feature level primitives are exported as a structured JSON file consumed by the NeuroHex world model builder and downstream applications.

Fig. 11: OSM2Hex conversion tool: (A) Raw OpenStreetMap data for the Pittsburgh metro area. (B) Pittsburgh metro area filtered and replaced with NeuroHex primitives. (C) Zoomed in Oakland area with higher resolution NeuroHex primitives.

Spatial Reasoning

Spatial reasoning studies how systems represent geometry, relations, viewpoints, and motion to understand scenes, plan actions, and solve spatial tasks.

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Abstract

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