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Ultimately, these are projections of [[Spatial model of voting|a multi-dimensional political space]] onto a space of fewer dimensions, to generalize and make discussion simpler.
===One-dimensional===
{{main|Left-right political spectrum}}
{{wikipedia|Left-wing politics}}
{{wikipedia|Right-wing politics}}
A single-dimensional model envisions a horizontal line, with voters distributed along a single left-to-right axis. This is frequently referred to as the [[Left-right political spectrum|left–right political spectrum]], and is how many people classify political positions, ideologies and parties. The people on the ends are said to practice [[extremism]], and the intermediate stance is called [[centrism]]. On this type of political spectrum, left-wing politics and right-wing politics are often presented as opposed, although a particular individual or group may take a left-wing stance on one matter and a right-wing stance on another; and some stances may overlap and be considered either left-wing or right-wing depending on the ideology.<ref>{{cite journal|last=Milner|first=Helen|date=2004|title=Partisanship, Trade Policy, and Globalization: Is There a Left–Right Divide on Trade Policy|url=https://www.princeton.edu/~hmilner/forthcoming%20papers/ISQ_milner_judkins2004.PDF|journal=International Studies Quarterly|volume=48|issue=|pages=95–120|doi=10.1111/j.0020-8833.2004.00293.x|pmid=|accessdate=|via=}}</ref> In [[France]], where the terms originated, the left has been called "the party of movement" and the right "the party of order".<ref>Knapp & Wright, p. 10.</ref><ref>Adam Garfinkle, Telltale Hearts: The Origins and Impact of the Vietnam Antiwar Movement (1997). Palgrave Macmillan: p. 303.</ref><ref>"[http://www.merriam-webster.com/dictionary/left Left (adjective)]" and "[http://www.merriam-webster.com/dictionary/left?show=1&t=1325146819 Left (noun)]" (2011), ''Merriam-Webster Dictionary''.</ref><ref>Roger Broad, ''Labour's European Dilemmas: From Bevin to Blair'' (2001). Palgrave Macmillan: p. xxvi.</ref>
Using the formulas above: n=1, v(x)=1, and d(x,y)=|x-y|. The directions on this spectrum are normally referred to as left and right.
==== Horseshoe theory ====
:''main article: [[horseshoe theory]]
What is often called the "[[horseshoe theory]]" claims that the extreme authoritarian economic left (Communism) is adjacent or close to extreme authoritarian economic right (neo-reactionism/fascism). A classification that follows this thought must then place these two close by or next to each other: either by using dimensions where they naturally fit next to each other, or by making opinion space curved so that going in the direction of fascism leads to Communism.
===Two-dimensional===
{{wikipedia|Nolan chart}}
{{wikipedia|The Poltical Compass}}
{{wikipedia|Pournelle chart}}
While the "horseshoe theory" appears two-dimensional, it is obviously just a variation on the [[left-right political spectrum]].
There are many two-dimensional political spaces, many of which have enough credible citations to have articles on [[English Wikipedia]]. These include the following:
* The [[W:Nolan chart]]
* [[W:The Political Compass]]
* The [[W:Pournelle chart]]
The ''Nolan chart'' and ''the Political Compass'' are two popular examples, which can be seen as rotated versions of each other. The ''Pournelle chart'' is another variation with a different set of axes. Other two-dimensional models are described below.
==== Three Telos Model ====
{{Main|Three Telos Model}}
[[File:Politics_map_triangle1.png|alt=|thumb]]
The "Three Telos Model" or "Triangle Political Map" is two-dimensional political model where voters tend to spread out in three directions. It describes political beliefs based on the core axiom of the philosophy, where the voter's depart from the center based on their core beliefs.
Each of the three colors (the "equality ''leftist''", the "freedom ''liberal"'' and the "tradition ''conservative''") have different criteria. The criteria are listed as:
* Justification
* Philosophical foundation
* Prestige idenifier
* Moral foundation<ref>Most people are sensitive to the fairness foundation</ref> (see [[Wikipedia:Moral foundations theory|Moral foundations theory]] on Wikipedia)
* Vision of nature <ref>[[W:A Conflict of Visions]]</ref><ref>{{Citation|title=Book sources|url=https://en.wikipedia.org/wiki/Special:BookSources/978-0688069124|work=Wikipedia|language=en|access-date=2021-01-14}}</ref><ref>{{Cite book|last=Sowell|first=Thomas|url=https://openlibrary.org/books/OL2724504M/A_conflict_of_visions|title=A conflict of visions|date=1987|publisher=W. Morrow|isbn=978-0-688-06912-4|edition=1st ed.|location=New York}}</ref> <ref>https://casnocha.com/2009/10/tragic-vs-utopian-view-of-human-nature.html</ref>
[[File:TelosTriangle.png|alt=|thumb|As in the two dimensional maps like the political compass, the differing ideologies can be put onto this map.]]
=== Three or higher dimensions ===
Political opinion can be divided into essentially any number of dimensions. Some other examples include the 3-dimensional [https://sapplyvalues.github.io Sapply Compass], the 4-dimensional [https://8values.github.io/ 8values] space, and the [https://9axes.github.io/ 9Axes] space.
One study of German voters found that at least four dimensions were required to adequately represent all political parties.<ref name=":2">{{Cite journal|last1=Alós-Ferrer|first1=Carlos|last2=Granić|first2=Đura-Georg|date=2015-09-01|title=Political space representations with approval data|url=http://repub.eur.nl/pub/111247|journal=Electoral Studies|volume=39|pages=56–71|doi=10.1016/j.electstud.2015.04.003|quote=The analysis reveals that the underlying political landscapes ... are inherently multidimensional and cannot be reduced to a single left-right dimension, or even to a two-dimensional space. ... From this representation, lower-dimensional projections can be considered which help with the visualization of the political space as resulting from an aggregation of voters' preferences. ... Even though the method aims to obtain a representation with as few dimensions as possible, we still obtain representations with four dimensions or more.|hdl=1765/111247}}</ref>
There has been references to many other political compasses that are similar, orthogonal or even contradictive.
* https://www.reddit.com/r/Politicaltests/wiki/listoftests (archive [https://archive.md/wip/amTxo https://archive.md/amTxo])
* https://l-lists.com/en/lists/0isll4.html (archive https://archive.md/YATlp)
* https://www.reddit.com/r/PoliticalCompass/comments/flzinl/list_of_every_test_i_know_including_some_you/ (https://archive.md/MOS9q)
In the end, it is difficult to model the behaviors of human beings in such a way that they can be reduced to simple numbers and political spectra as lines on a graph.
== Nonlinear spaces ==
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