2d ago

Goodfire AI researchers identify shape-rotating calculator in language models

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Goodfire AI researchers identified an internal mechanism in large language models that performs addition and related operations by rotating shapes in high-dimensional neural space. The reusable structure interfaces with Fourier features and reads from geometric representations learned during training. It extends to other computational tasks and builds on a 2023 arXiv paper on modular addition circuits. Released visualizations display shifting activation clusters in scatter plots.

Original post

Neural networks have beautiful feature geometry, but do they have mechanisms that actually interface with those structures? At @GoodfireAI this spring, we discovered one: a re-usable addition mechanism that reads/writes to Fourier features from prior work. 🧵

9:31 AM · May 14, 2026 View on X
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Very interesting finding! Reminds me of the structures Xiaoyan and Itamar found in multiplication reasoning:

GoodfireGoodfire@GoodfireAI

Neural networks do math by rotating shapes. We found a shape-rotating calculator hidden inside an LLM – and it’s used for more than just math! (1/6)

4:29 PM · May 14, 2026 · 786.7K Views
11:24 PM · May 14, 2026 · 10.1K Views

LLMs are wordcels AND shape rotators

GoodfireGoodfire@GoodfireAI

Neural networks do math by rotating shapes. We found a shape-rotating calculator hidden inside an LLM – and it’s used for more than just math! (1/6)

4:29 PM · May 14, 2026 · 786.7K Views
6:39 PM · May 14, 2026 · 25.7K Views

BREAKING: LLMs are just stochastic parrots

They have no idea of the real world. How could they? They are just storing text in their database and regurgitating it !!1

GoodfireGoodfire@GoodfireAI

Neural networks do math by rotating shapes. We found a shape-rotating calculator hidden inside an LLM – and it’s used for more than just math! (1/6)

4:29 PM · May 14, 2026 · 786.7K Views
7:12 PM · May 14, 2026 · 128.7K Views

(if you fell for this you are regarded)

Lisan al GaibLisan al Gaib@scaling01

BREAKING: LLMs are just stochastic parrots They have no idea of the real world. How could they? They are just storing text in their database and regurgitating it !!1

7:12 PM · May 14, 2026 · 128.7K Views
7:12 PM · May 14, 2026 · 4.2K Views
Lisan al GaibLisan al Gaib@scaling01

the stochastic parrot LLMs have obviously stolen this concept from mechanical calculators https://en.wikipedia.org/wiki/Mechanical_calculator

8:15 PM · May 14, 2026 · 9.6K Views
8:29 PM · May 14, 2026 · 5.7K Views

so cool. win for the shape rotators! @ericho_goodfire

GoodfireGoodfire@GoodfireAI

Neural networks do math by rotating shapes. We found a shape-rotating calculator hidden inside an LLM – and it’s used for more than just math! (1/6)

4:29 PM · May 14, 2026 · 786.7K Views
8:57 PM · May 14, 2026 · 3.3K Views

I figured we’d get here eventually.

It’s beautiful.

GoodfireGoodfire@GoodfireAI

Neural networks do math by rotating shapes. We found a shape-rotating calculator hidden inside an LLM – and it’s used for more than just math! (1/6)

4:29 PM · May 14, 2026 · 786.7K Views
5:32 PM · May 14, 2026 · 13.7K Views

the world inside neural networks is so beautiful. gradient descent learned to make a general-purpose addition module!

GoodfireGoodfire@GoodfireAI

Neural networks do math by rotating shapes. We found a shape-rotating calculator hidden inside an LLM – and it’s used for more than just math! (1/6)

4:29 PM · May 14, 2026 · 786.7K Views
4:31 PM · May 14, 2026 · 2.7K Views
Goodfire AI researchers identify shape-rotating calculator in language models · Digg