How Mathematics Improves The World | Making Digital Pictures Smaller Without Losing What Matters
A photograph can contain twenty-four million numbers and still look like one ordinary family dinner.
Take a 4000 × 3000 pixel colour image.
That is twelve million pixels.
If each pixel stores three eight-bit colour channels, the raw image contains roughly 288 million bits before metadata or formatting.
About 36 megabytes.
Yet a JPEG version of the same photograph may be two megabytes and look almost identical at ordinary viewing size.
Where did the missing 34 megabytes go?
They were not hidden.
They were recognised as structure that did not need to be represented in the most literal possible way.
Neighbouring pixels are correlated.
Human vision is more sensitive to some errors than others.
Smooth regions can be described compactly.
Fine high-frequency detail can sometimes be approximated without being noticed.
Repeated statistical patterns can be assigned short codes.
Image compression is the mathematics of deciding which information is essential, which is predictable, which is perceptually less important, and how all of it can be represented using fewer bits.
Quick Read
Digital images are large because every pixel carries numerical colour information. Compression reduces the number of bits needed to store or transmit that information. Lossless compression removes statistical redundancy while allowing exact reconstruction. Lossy compression goes further by allowing controlled changes that reduce data dramatically while preserving the aspects of the image judged most important for the intended use.
Classic JPEG is based on the discrete cosine transform, or DCT. The image is divided into small blocks, commonly 8 × 8 pixels. Each block is transformed from pixel values into a weighted sum of spatial frequency patterns. Smooth variation concentrates energy in a few low-frequency coefficients, while fine detail occupies higher-frequency coefficients. Quantisation then reduces precision—especially in higher frequencies where human vision often tolerates more error. Finally, entropy coding assigns compact binary codes to the resulting symbols.
The Joint Photographic Experts Group describes the core JPEG 1 system as a Huffman-coded DCT-based lossy format, while newer standards such as JPEG 2000 use wavelets and JPEG AI, published in 2025, uses an end-to-end learning-based image-coding approach.
The deep mathematical problem is rate–distortion: how few bits can represent an image while keeping distortion below an acceptable level? The answer depends on the viewer and the job. A holiday photo, medical image, satellite observation, fingerprint and computer-vision dataset may need different definitions of “what matters”.
One-sentence answer: Mathematics improves the world by discovering compact representations of visual information, allowing billions of images to be stored, transmitted and processed using far fewer bits while keeping the features that matter for human or machine use.
A Digital Picture Is a Matrix
Strip away the photograph.
What remains is an array of numbers.
A grayscale image can be represented as one matrix.
Each entry records brightness.
A colour image commonly has several channels.
Red, green and blue are one familiar representation.
JPEG commonly transforms colours into a luminance–chrominance space such as YCbCr because brightness and colour differences can be treated differently.
Once the picture becomes numbers, Mathematics can operate on it.
Add.
Transform.
Predict.
Quantise.
Encode.
A visual scene becomes an algebraic object.
The First Compression Opportunity: Neighbouring Pixels Are Similar
Photographs are not random noise.
A blue sky contains millions of neighbouring pixels with similar values.
A white wall changes slowly.
Even textured surfaces have local structure.
This means pixel values are statistically redundant.
If I tell you one sky pixel is bright blue, you can predict nearby pixels better than chance.
Compression exploits this predictability.
The central principle is:
Do not spend full information describing what the receiver could already predict.
Lossless Compression: Remove Redundancy, Keep Every Pixel
Lossless compression must reproduce the original data exactly.
Every pixel.
Every bit of sample data.
How can the file shrink if nothing is discarded?
By representing common patterns more efficiently.
Suppose an image row contains:
255, 255, 255, 255, 255, 255, 255, 255
Instead of writing eight copies, a run-length code can write:
255 repeated 8 times
Predictive coding can store differences from neighbouring pixels, which are often small and statistically concentrated.
Entropy coding assigns shorter bit strings to common symbols and longer bit strings to rare ones.
No information is lost.
The representation becomes less wasteful.
Entropy: Predictable Data Needs Fewer Bits
Claude Shannon’s information theory gives a lower bound on average code length for a source with known symbol probabilities.
If one symbol occurs almost all the time, the source carries little uncertainty.
If many symbols are equally likely, uncertainty is higher.
Compression removes coding inefficiency until the representation approaches the source’s information content.
This is why random noise compresses badly.
There is little statistical structure to exploit.
Compression is not magic.
It succeeds because real images are not maximally surprising.
Lossy Compression: Decide What Can Change
Lossless compression reaches a limit.
If every pixel must be reconstructed exactly, some information cannot be removed.
Lossy compression makes a different bargain.
Allow controlled error.
Then search for errors that are perceptually less important.
This is why lossy compression can reduce file size dramatically.
The codec is no longer asking:
How can I encode the exact image efficiently?
It asks:
Which approximate image gives the most useful visual quality for the available number of bits?
Rate–Distortion Theory: How Many Bits Is This Error Worth?
Rate is the number of bits used.
Distortion measures how different the reconstructed image is from the original.
Use more bits.
Distortion usually falls.
Use fewer bits.
Distortion rises.
The rate–distortion curve describes the best achievable trade-off under a chosen distortion measure.
The hidden difficulty is defining distortion.
Mean squared error is mathematically convenient.
Human visual perception does not experience all squared pixel differences equally.
A one-pixel edge shift can have large numerical error and look harmless.
A small structured artefact can have low average error and look terrible.
The metric defines what “good compression” means.
Human Vision Is Part of the Codec
Human eyes and brains are more sensitive to some image changes than others.
We are highly sensitive to edges and luminance structure.
We generally tolerate lower colour resolution better than lower brightness resolution.
Fine high-frequency texture can sometimes be removed with modest visual impact.
Compression formats exploit these properties.
This means a codec is partly a mathematical model of the receiver.
If the receiver is a human, visual perception matters.
If the receiver is a machine-learning model, a different distortion metric may be appropriate.
YCbCr: Separate Brightness From Colour Difference
RGB stores red, green and blue channels.
JPEG commonly converts to YCbCr.
Y represents luminance-like brightness.
Cb and Cr represent blue- and red-difference chroma components.
This transformation does not itself compress much.
It reorganises information into channels that match perceptual priorities better.
Once brightness and colour are separated, chroma can be sampled more coarsely.
The representation creates the opportunity for compression.
Chroma Subsampling: Store Less Colour Detail Than Brightness Detail
In 4:2:0 chroma subsampling, colour-difference information is stored at lower spatial resolution than luminance.
Four neighbouring luminance samples may share fewer chroma samples.
The file shrinks before the DCT has even started.
On photographs, the reduction is often difficult to notice.
On sharp coloured text, computer interfaces or synthetic graphics, chroma subsampling can create visible colour bleeding.
The same compression choice behaves differently on different content.
Good encoding is content-aware.
Why JPEG Uses 8 × 8 Blocks
Classic JPEG divides each image component into 8 × 8 blocks.
Why blocks?
The DCT is easier to compute on small fixed-size regions.
Local image statistics are easier to exploit.
Hardware implementations become practical.
Why eight?
It was an engineering compromise among compression performance, computational cost, memory and artefacts.
Small blocks adapt locally but create more boundaries.
Large blocks capture broader structure but cost more computation and can behave poorly near edges.
The standard chose a scale that worked well for the technology of its era and remains embedded in billions of devices.
The Discrete Cosine Transform: Rewrite Pixels as Frequencies
An 8 × 8 block contains 64 pixel values.
The DCT replaces those 64 values with 64 coefficients.
The number of values does not change.
Why is this useful?
Because the coefficients have different meanings.
One coefficient represents the average brightness of the block.
Others represent increasingly rapid horizontal and vertical variation.
A smooth patch of sky needs mostly low-frequency coefficients.
Many high-frequency coefficients are small.
The transform concentrates energy.
Compression becomes easier because important information moves into a small number of coefficients.
A Basis: Rebuild the Image From 64 Tiny Patterns
Each DCT coefficient multiplies a cosine basis pattern.
One pattern is constant.
Another varies slowly left to right.
Another varies slowly top to bottom.
Others alternate more rapidly.
Add all 64 weighted patterns and the original block returns exactly before quantisation, apart from numerical precision.
This is the same compositional Mathematics seen elsewhere.
Finite elements use shape functions.
Fourier series use sine and cosine waves.
Radiotherapy uses weighted beamlets.
Image compression uses spatial-frequency basis patterns.
Complex things become sums of simpler things.
The DCT Does Not Compress the Image Yet
This is a common misunderstanding.
The 64 pixel values become 64 DCT coefficients.
No information has been deliberately discarded.
The transform is invertible.
The benefit is that the new coordinates are better organised for compression.
The transform creates sparsity and unequal importance.
Quantisation does the destructive part.
Quantisation: The Moment Information Is Lost
Suppose one DCT coefficient is 127.
Divide by a quantisation step of 20.
Round to the nearest integer.
127 becomes 6.
During decompression, multiply back by 20.
6 becomes 120.
The lost seven units do not return.
This rounding is quantisation.
Large quantisation steps create many zeros and compress well.
They also create more distortion.
JPEG uses different quantisation strengths for different spatial frequencies, typically discarding more precision from high-frequency components.
The codec is deciding which numerical accuracy is worth paying for.
Why High Frequencies Are Often Cheaper to Lose
High-frequency DCT coefficients represent rapid pixel changes.
Fine texture.
Hair.
Grass.
Noise.
Sharp edges also contain high-frequency energy, which is why aggressive quantisation can damage them.
Human vision usually notices errors in coarse low-frequency structure more strongly than the loss of some fine texture.
So JPEG spends bits unevenly.
Not every coefficient deserves the same precision.
Zig-Zag Scanning: Put Likely Zeros Together
After quantisation, low-frequency coefficients tend to remain non-zero.
High-frequency coefficients often become zero.
JPEG orders coefficients in a zig-zag pattern from low to high frequency.
This tends to place long runs of zeros near the end.
Run-length coding can represent those zeros compactly.
A geometric ordering improves statistical coding.
Compression is a chain of small mathematical advantages.
Huffman Coding: Short Codes for Common Symbols
Once coefficients are converted into symbolic runs and values, entropy coding removes remaining redundancy.
Huffman coding assigns short bit patterns to common symbols and longer patterns to rare symbols while preserving unique decodability.
If “zero run of length 4” is common, it deserves a short code.
If an unusual large coefficient occurs rarely, it can afford a longer one.
This is Shannon’s principle turned into a practical tree.
Frequent events should cost fewer bits.
Arithmetic Coding: Encode an Entire Sequence as a Fraction
Arithmetic coding takes a different approach.
Instead of assigning an integer number of bits to each symbol independently, it represents a sequence as a subinterval of the number line.
More probable sequences receive larger intervals.
Less probable sequences receive smaller ones.
This can approach entropy limits more closely than simple Huffman coding for some probability distributions.
Classic JPEG includes arithmetic-coding options, though Huffman-coded JPEG became much more widespread historically.
Blocking Artefacts: The 8 × 8 Grid Reveals Itself
Compress a JPEG too aggressively and square blocks appear.
Why?
Each 8 × 8 block was transformed and quantised independently.
Neighbouring blocks may reconstruct their shared boundary differently.
At high quality, the mismatch is tiny.
At low quality, it becomes visible.
The artefact exposes the codec’s internal representation.
Every compression method leaves fingerprints of the assumptions it used.
Ringing: Sharp Edges Need Many Frequencies
A perfectly sharp edge contains high-frequency components.
Remove too many of them and the reconstructed edge develops ripples or halos.
This is related to the Gibbs phenomenon in truncated frequency representations.
The image may look oversharpened or surrounded by echoes.
Again, the artefact is mathematical.
The basis cannot represent a discontinuity cleanly after aggressive coefficient removal.
Quality Sliders Are Really Quantisation Controls
Many applications show a JPEG quality slider from 0 to 100.
That number is not standardised as one universal physical quality measure.
Different encoders map it to different quantisation tables and encoding decisions.
Quality 80 in one application is not guaranteed to equal quality 80 in another.
This is an important lesson for students.
An interface number can look universal while hiding implementation-specific meaning.
Always ask what the number actually controls.
Progressive JPEG: Let the Image Arrive in Layers
A baseline JPEG often decodes scan by scan in ordinary order.
Progressive JPEG can encode information in multiple passes.
The first pass produces a rough whole image.
Later passes refine detail.
This was especially useful on slow networks because users saw the image structure before every bit arrived.
The total information is reorganised according to usefulness over time.
Compression is not only about size.
It can also be about which information arrives first.
JPEG 2000: Replace Blocks With Wavelets
JPEG 2000 uses wavelet transforms rather than the block-based DCT architecture of classic JPEG.
Wavelets represent image structure at multiple spatial scales.
Coarse approximations describe broad image content.
Detail coefficients add horizontal, vertical and diagonal structure at finer scales.
This avoids classic 8 × 8 blocking artefacts and supports features such as quality scalability and region-of-interest coding.
NIST describes JPEG 2000 succinctly as a wavelet-based image compression and storage format.
Different basis.
Same deep idea:
transform the image into a representation where important structure can be separated from less important detail.
Wavelets: Localised Frequencies
Fourier sinusoids extend across an entire signal.
Wavelets are localised in both scale and position.
This makes them well suited to images where edges are local features.
A wavelet decomposition produces coefficients representing:
- large-scale approximation;
- horizontal detail;
- vertical detail;
- diagonal detail;
- and finer versions recursively.
Many small coefficients can be quantised heavily or discarded.
Important edges survive through larger coefficients.
Multiscale representation becomes compression.
Compression and Error Correction Solve Opposite Problems
In Making Noisy Messages Arrive Correctly, we deliberately added redundancy.
In compression, we remove redundancy.
Why are both useful?
Because the redundancy is different.
Compression removes predictable structure already present in the source.
Error correction adds carefully designed structure that protects against channel corruption.
A communication system often compresses first, then error-protects the compressed stream.
The same bit can be waste in one layer and safety in another.
Repeated JPEG Saves: Quantisation Damage Accumulates
Open a JPEG.
Edit it.
Save as JPEG again.
Each lossy save may re-transform and re-quantise the image.
Errors accumulate.
Fine detail degrades.
Blocking and ringing can worsen.
This is why professional editing workflows preserve high-quality or lossless masters and export lossy delivery copies only when needed.
Compression is not reversible once quantisation has thrown information away.
Medical Images: “Looks Fine” Is Not a Sufficient Distortion Metric
For a social-media photograph, a tiny texture loss may be irrelevant.
For a diagnostic image, subtle detail may matter clinically.
Lossy compression therefore requires domain-specific validation.
A codec optimised for average human preference may erase a small feature important to a radiologist.
This is a powerful general lesson.
Distortion is not a universal property.
It is distortion relative to a task.
What matters depends on who—or what—is looking.
Machine Vision Changes What “Important” Means
Suppose an image is not meant for a human.
It is meant for an object detector.
Or a face-recognition model.
Or a satellite segmentation algorithm.
A compression artefact invisible to a person may reduce model accuracy.
Conversely, a change a person notices may not affect the machine’s task.
Modern image coding therefore increasingly considers machine-oriented metrics as well as human visual quality.
The JPEG AI standard explicitly targets both human visualisation and effective performance for image-processing and computer-vision tasks.
JPEG AI: Let a Neural Network Learn the Representation
Classic JPEG uses a hand-designed transform.
JPEG AI uses learned analysis and synthesis transforms trained on large image datasets.
An encoder neural network maps the image into a compact latent representation.
That latent representation is quantised and entropy-coded.
A decoder network reconstructs the image.
The network learns basis-like features suited to natural images rather than relying only on fixed cosine patterns.
The JPEG Committee reports that Version 1 of JPEG AI was published in 2025 as the first international image-coding standard based on an end-to-end learning-based approach.
The mathematics did not disappear.
It moved into optimisation, probability, neural networks and learned transforms.
The Learned Codec Has New Failure Modes
A hand-designed DCT is predictable.
A learned codec can adapt more richly to image statistics.
But it also inherits training data.
Rare image classes may reconstruct poorly.
Unusual scientific textures may be treated as noise.
Adversarial or out-of-distribution images may behave unexpectedly.
Perceptual losses can favour plausible-looking detail that is not pixel-faithful.
That last failure is critical in evidence-bearing images.
A beautiful reconstruction must not invent information and then present it as observation.
Perceptual Compression: When Plausible Is Better Than Exact—Until It Is Not
At extremely low bitrates, pixel-perfect reconstruction is impossible.
Generative models can produce images that look more natural by replacing lost detail with statistically plausible texture.
For entertainment imagery, this may be desirable.
For scientific or forensic imagery, invented texture can be unacceptable.
This creates a fundamental three-way tension:
- bitrate;
- perceptual quality;
- fidelity to the exact source.
You cannot optimise all three without limit.
The correct trade-off depends on the use.
Compression Enables the Modern Internet
Imagine every photograph uploaded at raw camera size.
Every website would load more slowly.
Storage systems would need far more capacity.
Mobile data costs would rise.
Data centres would use more energy.
Satellite links would carry fewer images.
Digital archives would cost more to preserve.
Compression converts mathematical insight directly into infrastructure capacity.
A 10× compression ratio acts like 10× more storage or 10× more transmission capacity for the same class of visual data, ignoring overheads and quality trade-offs.
Better representation can substitute for physical resources.
Energy: Fewer Bits Also Mean Fewer Joules
Moving data consumes energy.
Storing data consumes energy.
Processing compression also consumes energy.
So the best codec depends on the whole energy balance.
A very complex encoder may save bandwidth but cost heavy computation.
That can be acceptable in a cloud service encoding once and serving millions of copies.
It may be poor for a tiny battery-powered camera.
Compression efficiency is not only file size.
It is system efficiency.
Latency: A Smaller File Can Take Longer to Create
A sophisticated codec may compress better and encode slower.
For an archive, that may be fine.
For a live camera preview, it may be unusable.
Decoding complexity matters too.
A billion smartphones must decode images quickly with limited battery.
Standards therefore negotiate among:
- compression ratio;
- quality;
- encoding speed;
- decoding speed;
- memory;
- hardware complexity;
- licensing;
- backward compatibility.
The mathematically smallest file is not automatically the best standard.
Standards: Compression Works Because Everybody Agrees How to Undo It
A private compression algorithm is useful only if the receiver has the matching decoder.
JPEG became world-changing because it became a standard.
A camera made by one company can write a file.
A browser from another can read it.
A printer from a third can decode it.
A standards organisation turns Mathematics into interoperability.
The compression algorithm solves file size.
The standard solves social coordination.
Metadata: The Pixels Are Not the Whole Image File
A JPEG file can contain metadata.
Camera model.
Date.
Orientation.
Colour profile.
Sometimes GPS location.
Thumbnail.
Compression reduces the pixel payload.
It does not automatically protect privacy.
A heavily compressed holiday photo can still reveal where it was taken through metadata.
File size and information privacy are different problems.
Forensics: Compression Leaves Traces
JPEG block structure and quantisation tables can leave detectable statistical patterns.
Double compression can sometimes be inferred.
Edited regions may have inconsistent compression history.
Image forensics uses these traces as evidence.
But forensic inference is probabilistic.
Many legitimate processing pipelines produce similar artefacts.
A compression fingerprint is evidence, not automatic proof of manipulation.
Adversarial Images: Compression Can Help or Hurt Machine Learning
Machine-learning systems can be sensitive to tiny pixel perturbations.
JPEG compression may remove some high-frequency adversarial noise.
But it can also create artefacts that change model outputs.
An attacker can design perturbations that survive compression.
The relationship among compression, perception and machine inference is still an active research area.
The broader lesson is clear:
an image representation that is good for a human is not guaranteed to be safe for an algorithm.
A Classroom Thought Experiment: Which Numbers Can You Throw Away?
Write sixteen numbers representing a 4 × 4 grayscale patch:
100 101 100 102
99 100 101 101
100 100 99 101
101 100 100 100
Ask the student what the patch looks like.
Almost uniform grey.
Could we replace all sixteen values with one average of 100?
We would lose information.
The patch would probably look nearly the same.
Now create another patch with a sharp black-and-white edge.
Replacing it with the average destroys the important structure.
The student discovers the essence of compression:
which differences matter depends on the structure.
A Second Thought Experiment: Rebuild a Picture From Basis Tiles
Print simple black-and-white patterns.
Constant grey.
A gradual left-to-right gradient.
A top-to-bottom gradient.
Alternating stripes.
Explain that a DCT block is represented as weighted combinations of patterns like these.
A smooth image needs large weights on simple patterns.
A highly textured image needs many complex patterns.
The student learns basis decomposition without needing the cosine formula yet.
Primary Mathematics: Compression Begins With Patterns
Primary students already learn the foundations:
- patterns;
- averages;
- tables;
- coordinates;
- fractions;
- ratios;
- rounding;
- place value.
Rounding is lossy compression in miniature.
3.14159265 becomes 3.14.
Information is lost.
The approximation may be good enough for the task.
The child already understands rate–distortion intuitively:
how much precision do we need?
Secondary Mathematics: Images Become Transformations
Secondary students add:
- matrices;
- vectors;
- trigonometric functions;
- probability;
- logarithms;
- statistics;
- functions.
A matrix stores pixels.
A linear transform changes basis.
Cosines form the DCT basis.
Probability models symbol frequencies.
Logarithms measure information in bits.
School Mathematics becomes a camera format.
Advanced Mathematics: Where Compression Lives
Image compression draws on:
- linear algebra;
- Fourier analysis;
- wavelets;
- probability;
- information theory;
- optimisation;
- quantisation theory;
- signal processing;
- statistics;
- machine learning.
Transforms concentrate energy.
Quantisation spends precision selectively.
Entropy coding removes statistical redundancy.
Rate–distortion theory defines the trade-off.
Perceptual models define what distortion matters.
Neural networks learn representations directly from data.
What Mathematics Improves Here
1. It reduces storage
Compressed images require far less disk and cloud capacity than raw pixel arrays.
2. It reduces transmission time
Smaller files travel faster across mobile, satellite and internet links.
3. It makes visual communication scalable
Billions of photos can be shared daily only because visual information is represented efficiently.
4. It allocates bits according to importance
Transforms and perceptual models spend more bits on structure viewers notice and fewer on redundant or less visible detail.
5. It creates interoperable visual standards
Standardised codecs let devices from different companies exchange images reliably.
6. It reveals that representation itself is a resource
Better representation can create the equivalent of more bandwidth, more storage and lower energy use without changing the underlying physical hardware.
What Mathematics Does Not Do
Compression does not create information that was never captured.
Lossy compression does not preserve every pixel.
A high visual-quality score does not guarantee medical or scientific fidelity.
A smaller file is not automatically better.
A learned codec does not guarantee it will not hallucinate plausible texture under extreme compression.
Compression does not protect privacy or authenticity.
And a codec optimised for humans is not guaranteed to preserve the information a machine vision system needs.
Frequently Asked Questions
What is JPEG?
JPEG commonly refers to the still-image coding standard ISO/IEC 10918 / ITU-T T.81 created by the Joint Photographic Experts Group. Its widely used lossy mode is based on 8 × 8 discrete cosine transforms, quantisation and entropy coding.
Does the DCT itself lose information?
In principle, the DCT is an invertible linear transform. The deliberate information loss in ordinary JPEG occurs mainly during quantisation, where transformed coefficients are rounded to coarser values.
Why do JPEG images become blocky?
Because classic JPEG processes 8 × 8 blocks independently. Heavy quantisation can make neighbouring blocks reconstruct differently at their boundaries, revealing the hidden grid.
What is chroma subsampling?
Chroma subsampling stores colour-difference information at lower spatial resolution than brightness information, exploiting the fact that human vision often tolerates reduced chroma detail better than reduced luminance detail.
What is JPEG 2000?
JPEG 2000 is a later image-compression standard based on wavelet transforms rather than the block-DCT architecture of classic JPEG. It supports scalable decoding, lossless modes and other advanced features.
What is JPEG AI?
JPEG AI is the first international image-coding standard based on an end-to-end learning-based approach. Version 1 was published in 2025 and targets efficient image coding for human visualisation as well as machine-processing tasks.
Sources and Further Reading
- Joint Photographic Experts Group, JPEG 1, describing the core DCT-based lossy coding system and related JPEG standards.
- NIST, JPEG, identifying JPEG as a DCT-based image-compression and storage format.
- NIST, JPEG 2000, identifying the standard as wavelet based.
- Joint Photographic Experts Group, JPEG AI, describing the learning-based ISO/IEC 6048 / ITU-T T.840 image-coding standard published in 2025.
- Claude E. Shannon, A Mathematical Theory of Communication, 1948, foundational to entropy and rate–distortion thinking.
Continue Through eduKateSG
Continue with How Mathematics Works. Compare this article with Making Noisy Messages Arrive Correctly: compression removes predictable redundancy while channel coding adds carefully designed redundancy. Together they make digital communication efficient and resilient.
Final Thought: A Smaller Picture Can Still Contain the Same Moment
A photograph is not its pixels.
At least, not to the person looking at it.
It is a face.
A sunset.
A classroom.
A birthday.
A piece of evidence.
A scientific observation.
Mathematics asks which numerical details carry that meaning and which are merely one expensive way of storing it.
Then it changes coordinates.
Rounds carefully.
Codes common patterns efficiently.
And leaves us with a file small enough to travel across the world in a second.
The picture became smaller.
The moment did not have to.