The texture problem
The PlayStation could not perform perspective-correct texture mapping. It used affine texture mapping instead, interpolating UV coordinates between vertices using only their two-dimensional screen positions and ignoring each vertex's depth. If a triangle's vertices sit at different distances from the camera — which, on any surface not exactly parallel to the screen, they do — the texture is stretched linearly across the flat triangle with no regard for perspective, and it visibly swims as the camera moves.
The reason is arithmetic the console could not pay for. Perspective correction requires dividing each pixel by its depth value, and the PS1's GPU did not retain z-values after rasterisation, so the information was not there to divide by. Even if it had been, division is expensive and doing it per pixel would have been far too slow. This is not an oversight; it is a deliberate omission by a design team choosing what to spend transistors and cycles on, and texture perspective lost.
The vertex problem
The wobble people describe is often not the textures at all but the geometry, and that has a different cause entirely. The PlayStation's rasteriser works only with integer coordinates — it does not accept vertex positions with subpixel precision. Every vertex is therefore rounded to the nearest whole pixel before being drawn, which means that as an object moves smoothly through space, its corners snap from pixel to pixel rather than sliding.
The effect is most visible on large polygons at middle distance, and on anything moving slowly: edges shiver, and a character's face seems to ripple as vertices independently round up or down between frames. This is a fixed-point precision limitation, entirely separate from the texture issue. A game could in principle exhibit one without the other. That both appear together on nearly every PS1 title, producing a single compound impression of instability, is why they get discussed as one phenomenon.
Working around it
Developers found the same mitigation for the texture problem that the mathematics suggests: subdivide. Affine mapping's error grows with the depth disparity across a triangle, so smaller triangles have less error. Later PS1 games use more, smaller polygons, which means the warping happens across each little triangle rather than across a whole wall, and becomes proportionally less noticeable. It does not fix anything — the error is still there — it just distributes it below the threshold where the eye reliably catches it.
That trade has a cost, since more triangles is exactly what the hardware could least afford, and it explains a visible progression across the console's lifetime. Early PS1 games have large, cleanly-textured surfaces that swim alarmingly; later ones have busier geometry that holds together better. The improvement is not the developers getting better at 3D in general. It is them spending their polygon budget buying down a specific defect.
The defect as signature
What makes this worth writing about is where it ended up. The PS1's wobble was, unambiguously, a failure — the console was doing 3D wrong in two independent ways, and every developer who noticed spent resources hiding it. Sony's competitors did it better; the Nintendo 64 had perspective-correct texture mapping and subpixel precision, and its geometry sits still.
And yet the wobble is now a deliberate aesthetic. Modern games and shaders reproduce affine texture mapping and vertex snapping on purpose, in engines that would render correctly by default, because that specific instability reads as "PlayStation" to anyone who was there. Developers write code to reintroduce a division they are perfectly capable of performing, and to throw away precision their hardware has in abundance, in order to recreate the exact appearance of a machine failing at its job. The N64 got the mathematics right and has no comparable signature. Being wrong in a distinctive way turned out to be more memorable than being right.