User:Quickfur (Meta, no ontology)

Projections are the best way to understand higher dimensions. Our eyes see only in 2D, yet we unconsciously infer 3D from the 2D images. Similarly, a 4D being would see in only in 3D, and infer 4D from it. Since we are familiar with 3D, it is not difficult to understand what the 4D being sees in its retina. From there, we just have to deduce 4D depth in order to "see" in 4D.

By applying dimensional analogy to such 3D projections of 4D objects, we can ''intuitively'' derive many properties of 4-space that are otherwise obtained only through painstaking mathematical analysis and deduction.

==4D shapes of interest==

(This list should be merged into the main page eventually.)

* Polychora:
:* Regular polychora:
::* [[Pentachoron]]
::* [[Tesseract]]
::* [[16-cell]]
::* [[24-cell]]
::* [[120-cell]]
::* [[600-cell]]
:* [[Table_of_uniform_polychora_by_Bowers_acronym|Uniform polychora]]:
::* Pentachoron family:
::* Tesseract/16-cell family:
::* 120-cell/600-cell family:
::* The [[grand antiprism]]: an anomalous, non-Wythoffian uniform polychoron
::* The [[polyhedral prism]]s
::* The [[duoprism]]s
:* The [[Catalan polychora]]
* Rotachora:
:* Spherical:
::* [[Spherinder]] - sphere prism
::* [[Sphone]] - sphere pyramid
:* Prismic:
::* [[Cubinder]] - cylinder prism; Cartesian product of circle and square
::* [[Coninder]] - cone prism
:* Duo-cyclic:
::* [[Duocylinder]] - Cartesian product of two circles
::* [[Prismic cylinders]] (polygon-circle Cartesian products)

==Notes on derivation of uniform polychora==

The uniform polychora may be constructed from the regular polychora by various operations:

'''Truncation''' of vertices to different levels yields the truncate, the rectate, the bitruncate, the dual's rectate, the dual's truncate, and the dual itself. If the polychoron is self-dual, only half of this list will be unique, and the bitruncate will be cell-transitive.

'''Cantellation''' (truncation of vertices and edges) yields cantellates.

'''Runcination''' (truncation of vertices, edges, and faces) yields runcinates, identical for dual polychora.

'''Omnitruncation''' (truncation, cantellation, and runcination combined) yields omnitruncates, also identical for dual polychora.

'''Diminishing''' yields the [[grand antiprism]] and the [[snub 24-cell]] from the 600-cell; however, there is no single diminishing operation that can consistently derive these polychora (the particular diminishings that yield the grand antiprism and the snub 24-cell are case-specific).

==Johnson polychora discovery project==

[[CRF polychora discovery project]]: a project to catalogue all 4D Johnson polychora (defined as all convex 4D polychora having regular polygons for ridges).