← Mathematical compendium

Published equation contexts

Ntok(H,W)=⌈HP⌉×⌈WP⌉N_{\mathrm{tok}}(H,W) = \left\lceil \frac{H}{P} \right\rceil \times \left\lceil \frac{W}{P} \right\rceil

Why this formula appears here

A single formula covers all three counting schemes if the tile or patch edge length is left as a free constant P , specific to the vendor: Ntok(H,W)=⌈HP⌉×⌈WP⌉N_{\mathrm{tok}}(H,W) = \left\lceil \frac{H}{P} \right\rceil \times \left\lceil \frac{W}{P} \right\rceil. with a request’s image cost simply costimage\mathrm{cost}_{\mathrm{image}} = Ntok(H,W)N_{\mathrm{tok}}(H,W) ⋅\cdot ptokp_{\mathrm{tok}} at the model’s own per-token price ptokp_{\mathrm{tok}} . The constant differs — 28 pixels for Claude, 32 for GPT-5.4, roughly 768 divided into geometry-dependent tiles for Gemini — but the shape does not: token count, and therefore cost, scales with the area of the image, not its linear size. Doubling both width and height quadruples the token bill under every one of these three schemes. That is the single fact behind…

Read the full article-specific guide →

Read the representative guide

NtokN_{\mathrm{tok}}

Symbol N_tok

NtN_tok is part of the quantity the equation computes from the expression on the right.

Read this term in its guide →
PP

Symbol P

P occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Read this term in its guide →

How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

Research cited beside this formula

Learn the mathematics behind it

Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

Ntok(H,W)=⌈HP⌉×⌈WP⌉N_{\mathrm{tok}}(H,W) = \left\lceil \frac{H}{P} \right\rceil \times \left\lceil \frac{W}{P} \right\rceil

Equation 4 · Foundation Models

What Multimodal AI Actually Costs, Modality by Modality

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

A single formula covers all three counting schemes if the tile or patch edge length is left as a free constant P , specific to the vendor: Ntok(H,W)=⌈HP⌉×⌈WP⌉N_{\mathrm{tok}}(H,W) = \left\lceil \frac{H}{P} \right\rceil \times \left\lceil \frac{W}{P} \right\rceil. with a request’s image cost simply costimage\mathrm{cost}_{\mathrm{image}} = Ntok(H,W)N_{\mathrm{tok}}(H,W) ⋅\cdot ptokp_{\mathrm{tok}} at the model’s own per-token price ptokp_{\mathrm{tok}} . The constant differs — 28 pixels for Claude, 32 for GPT-5.4, roughly 768 divided into geometry-dependent tiles for Gemini — but the shape does not: token count, and therefore cost, scales with the area of the image, not its linear size. Doubling both width and height quadruples the token bill under every one of these three schemes. That is the single fact behind…

Equation guide → · Article →