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The conjugate groupings of uniform polyhedra in Tito Pieza III's list correspond to abstract isomorphism classes of uniform polyhedra except that:

  • 13 and 14 in his list form one abstract isomorphism class ($U_{33},U_{61},U_{43},U_{42}$ are all abstractly isomorphic),
  • 23 ($U_{46}$) is actually abstractly isomorphic to $U_{64}$,
  • $U_{30}$ and $U_{47}$ are abstractly isomorphic, but not on the list of conjugates,
  • $U_{62}$ and $U_{65}$ are also abstractly isomorphic and not on the list of conjugates,
  • and finally, 22 ($U_{12}$) is in an abstract isomorphism class on its own, as are $U_n$ with $n=1, 2,3,4,5,6,7,8,15,16,36,38,41,44,45,56,59,62,65,75$ (the same list as in his comment on that answer, except that 30, 47, 62, 64, 65 are omitted, per above).

(Here's the text document I compiled while searching Klitzing's site that includes the names of the polyhedra in nonsingleton abstract isomorphism classes and links.)

edit: Klitzing claims on his pages (as of 18 Feb 2021) that $U_{33},U_{61},U_{43},U_{42}$ are all abstractly isomorphic (see for example this archive link). In earlier versions of this answer, I repeated that claim here, stating that the two conjugate groupings 13 ($U_{42}$ and $U_{43}$) and 14 ($U_{33}$ and $U_{61}$) formed one abstract isomorphism class. However, Daniel Sebald pointed out in a comment on this answer that this isn't true; the 1-skeleta of $U_{33}$ and $U_{61}$ contain squares and those of $U_{42}$ and $U_{43}$ do not!

The conjugate groupings of uniform polyhedra in Tito Pieza III's list correspond to abstract isomorphism classes of uniform polyhedra except that:

  • 13 and 14 in his list form one abstract isomorphism class ($U_{33},U_{61},U_{43},U_{42}$ are all abstractly isomorphic),
  • 23 ($U_{46}$) is actually abstractly isomorphic to $U_{64}$,
  • $U_{30}$ and $U_{47}$ are abstractly isomorphic, but not on the list of conjugates,
  • $U_{62}$ and $U_{65}$ are also abstractly isomorphic and not on the list of conjugates,
  • and finally, 22 ($U_{12}$) is in an abstract isomorphism class on its own, as are $U_n$ with $n=1, 2,3,4,5,6,7,8,15,16,36,38,41,44,45,56,59,62,65,75$ (the same list as in his comment on that answer, except that 30, 47, 62, 64, 65 are omitted, per above).

(Here's the text document I compiled while searching Klitzing's site that includes the names of the polyhedra in nonsingleton abstract isomorphism classes and links.)

The conjugate groupings of uniform polyhedra in Tito Pieza III's list correspond to abstract isomorphism classes of uniform polyhedra except that:

  • 23 ($U_{46}$) is actually abstractly isomorphic to $U_{64}$,
  • $U_{30}$ and $U_{47}$ are abstractly isomorphic, but not on the list of conjugates,
  • $U_{62}$ and $U_{65}$ are also abstractly isomorphic and not on the list of conjugates,
  • and finally, 22 ($U_{12}$) is in an abstract isomorphism class on its own, as are $U_n$ with $n=1, 2,3,4,5,6,7,8,15,16,36,38,41,44,45,56,59,62,65,75$ (the same list as in his comment on that answer, except that 30, 47, 62, 64, 65 are omitted, per above).

(Here's the text document I compiled while searching Klitzing's site that includes the names of the polyhedra in nonsingleton abstract isomorphism classes and links.)

edit: Klitzing claims on his pages (as of 18 Feb 2021) that $U_{33},U_{61},U_{43},U_{42}$ are all abstractly isomorphic (see for example this archive link). In earlier versions of this answer, I repeated that claim here, stating that the two conjugate groupings 13 ($U_{42}$ and $U_{43}$) and 14 ($U_{33}$ and $U_{61}$) formed one abstract isomorphism class. However, Daniel Sebald pointed out in a comment on this answer that this isn't true; the 1-skeleta of $U_{33}$ and $U_{61}$ contain squares and those of $U_{42}$ and $U_{43}$ do not!

added 8 characters in body
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j.c.
j.c.
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Igor Pak's answer suggests that conjugate polyhedra in the sense of the question should be different realizations of the same abstract polyhedron (defined by a face lattice); so in this answer I'll list those which are abstractly isomorphic. The two main references for this answer are the paper "Closed-Form Expressions for Uniform Polyhedra and Their Duals" by Peter Messer and the page "Polytopes & their Incidence Matrices" by Richard Klitzing.

Abstract isomorphism

(In earlier revisions of this answer I attempted to say something about abstract isomorphism from the table of vertex cycle symbols in Messer's paper, but as I wasn't able to get very far with that approach, I've deleted it. The first two comments to this answer refer to that text.)

Klitzing's pagesite seems to be unique in that theythe subpages indicate which sets of polyhedra are abstractly isomorphic, but it's unfortunately a bit tricky to navigate (I could not find an easy to parse list of just the 75 non-prismatic uniform polytopes; the closest was this). With Wikipedia's list of uniform polyhedra, I managed to find each of the polyhedra on his site by googling their names with the keyword "site:bendwavy.org".

Here's a summary:

The conjugate groupings of uniform polyhedra in Tito Pieza III's list correspond to abstract isomorphism classes of uniform polyhedra except that:

  • 13 and 14 in his list form one abstract isomorphism class ($U_{33},U_{61},U_{43},U_{42}$ are all abstractly isomorphic),
  • 23 ($U_{46}$) is actually abstractly isomorphic to $U_{64}$,
  • $U_{30}$ and $U_{47}$ are abstractly isomorphic, but not on the list of conjugates,
  • $U_{62}$ and $U_{65}$ are also abstractly isomorphic and not on the list of conjugates,
  • and finally, 22 ($U_{12}$) is in an abstract isomorphism class on its own, as are $U_n$ with $n=1, 2,3,4,5,6,7,8,15,16,36,38,41,44,45,56,59,62,65,75$ (the same list as in his comment on that answer, except that 30, 47, 62, 64, 65 are omitted, per above).

(Here's the text document I compiled while searching Klitzing's site that includes the names of the polyhedra in nonsingleton abstract isomorphism classes and links.)

Circumradii

To get from this to the squared circumradii, the results in the paper cited in Igor Pak's answer state that the squared circumradii of two abstractly isomorphic polyhedra should be roots of the same polynomial (actually we require some generalization of those theorems to non-simplicial polyhedra, which might be straightforward).

One thing to keep in mind is that the polynomial equations satisfied by general polynomial invariants (in the sense of Fedorchuk and Pak's paper) could be reducible. So their results alone are not sufficient to show that circumradii of uniform polyhedra that are abstractly isomorphic are actually Galois conjugate.

Indeed, the fact that the abstract isomorphism classes above don't all correspond to the Galois conjugacy classes found previously bear this out!

This is of course a good conceptual framework to think about what's going on, but a more hands-on approach is also possible:

Messer gives explicit expressions for the circumradii in his paper; in section 8, the circumradius ${}_{0}R$ is expressed as $\csc\phi$, where $\phi$ is the angle subtended by a half-edge when viewed from the center of the circumsphere. In section 9, he gives expressions for $\cot\phi$ for the uniform polyhedra for the first 3 of the 4 "forms" mentioned in Tito Pieza III's answer. Appendix A contains polynomials that define $Y=2-\tan^2\phi$ for the fourth "snub" case, and in fact the table given there places the Wythoff symbols together when their $Y$ satisfy the same polynomial. In each case, it should be possible to check the conjugacy from these expressions (after applying some trigonometric identities and rewriting certain cosines of rational multiples of $\pi$ as roots of polynomials).

Igor Pak's answer suggests that conjugate polyhedra in the sense of the question should be different realizations of the same abstract polyhedron (defined by a face lattice); so in this answer I'll list those which are abstractly isomorphic. The two main references for this answer are the paper "Closed-Form Expressions for Uniform Polyhedra and Their Duals" by Peter Messer and the page "Polytopes & their Incidence Matrices" by Richard Klitzing.

Abstract isomorphism

(In earlier revisions of this answer I attempted to say something about abstract isomorphism from the table of vertex cycle symbols in Messer's paper, but as I wasn't able to get very far with that approach, I've deleted it. The first two comments to this answer refer to that text.)

Klitzing's page seems to be unique in that they indicate which sets of polyhedra are abstractly isomorphic, but it's unfortunately a bit tricky to navigate (I could not find an easy to parse list of just the 75 non-prismatic uniform polytopes; the closest was this). With Wikipedia's list of uniform polyhedra, I managed to find each of the polyhedra on his site by googling their names with the keyword "site:bendwavy.org".

Here's a summary:

The conjugate groupings of uniform polyhedra in Tito Pieza III's list correspond to abstract isomorphism classes of uniform polyhedra except that:

  • 13 and 14 in his list form one abstract isomorphism class ($U_{33},U_{61},U_{43},U_{42}$ are all abstractly isomorphic),
  • 23 ($U_{46}$) is actually abstractly isomorphic to $U_{64}$,
  • $U_{30}$ and $U_{47}$ are abstractly isomorphic, but not on the list of conjugates,
  • $U_{62}$ and $U_{65}$ are also abstractly isomorphic and not on the list of conjugates,
  • and finally, 22 ($U_{12}$) is in an abstract isomorphism class on its own, as are $U_n$ with $n=1, 2,3,4,5,6,7,8,15,16,36,38,41,44,45,56,59,62,65,75$ (the same list as in his comment on that answer, except that 30, 47, 62, 64, 65 are omitted, per above).

(Here's the text document I compiled while searching Klitzing's site that includes the names of the polyhedra in nonsingleton abstract isomorphism classes and links.)

Circumradii

To get from this to the squared circumradii, the results in the paper cited in Igor Pak's answer state that the squared circumradii of two abstractly isomorphic polyhedra should be roots of the same polynomial (actually we require some generalization of those theorems to non-simplicial polyhedra, which might be straightforward).

One thing to keep in mind is that the polynomial equations satisfied by general polynomial invariants (in the sense of Fedorchuk and Pak's paper) could be reducible. So their results alone are not sufficient to show that circumradii of uniform polyhedra that are abstractly isomorphic are actually Galois conjugate.

Indeed, the fact that the abstract isomorphism classes above don't all correspond to the Galois conjugacy classes found previously bear this out!

This is of course a good conceptual framework to think about what's going on, but a more hands-on approach is also possible:

Messer gives explicit expressions for the circumradii in his paper; in section 8, the circumradius ${}_{0}R$ is expressed as $\csc\phi$, where $\phi$ is the angle subtended by a half-edge when viewed from the center of the circumsphere. In section 9, he gives expressions for $\cot\phi$ for the uniform polyhedra for the first 3 of the 4 "forms" mentioned in Tito Pieza III's answer. Appendix A contains polynomials that define $Y=2-\tan^2\phi$ for the fourth "snub" case, and in fact the table given there places the Wythoff symbols together when their $Y$ satisfy the same polynomial. In each case, it should be possible to check the conjugacy from these expressions (after applying some trigonometric identities and rewriting certain cosines of rational multiples of $\pi$ as roots of polynomials).

Igor Pak's answer suggests that conjugate polyhedra in the sense of the question should be different realizations of the same abstract polyhedron (defined by a face lattice); so in this answer I'll list those which are abstractly isomorphic. The two main references for this answer are the paper "Closed-Form Expressions for Uniform Polyhedra and Their Duals" by Peter Messer and the page "Polytopes & their Incidence Matrices" by Richard Klitzing.

Abstract isomorphism

(In earlier revisions of this answer I attempted to say something about abstract isomorphism from the table of vertex cycle symbols in Messer's paper, but as I wasn't able to get very far with that approach, I've deleted it. The first two comments to this answer refer to that text.)

Klitzing's site seems to be unique in that the subpages indicate which sets of polyhedra are abstractly isomorphic, but it's unfortunately a bit tricky to navigate (I could not find an easy to parse list of just the 75 non-prismatic uniform polytopes; the closest was this). With Wikipedia's list of uniform polyhedra, I managed to find each of the polyhedra on his site by googling their names with the keyword "site:bendwavy.org".

Here's a summary:

The conjugate groupings of uniform polyhedra in Tito Pieza III's list correspond to abstract isomorphism classes of uniform polyhedra except that:

  • 13 and 14 in his list form one abstract isomorphism class ($U_{33},U_{61},U_{43},U_{42}$ are all abstractly isomorphic),
  • 23 ($U_{46}$) is actually abstractly isomorphic to $U_{64}$,
  • $U_{30}$ and $U_{47}$ are abstractly isomorphic, but not on the list of conjugates,
  • $U_{62}$ and $U_{65}$ are also abstractly isomorphic and not on the list of conjugates,
  • and finally, 22 ($U_{12}$) is in an abstract isomorphism class on its own, as are $U_n$ with $n=1, 2,3,4,5,6,7,8,15,16,36,38,41,44,45,56,59,62,65,75$ (the same list as in his comment on that answer, except that 30, 47, 62, 64, 65 are omitted, per above).

(Here's the text document I compiled while searching Klitzing's site that includes the names of the polyhedra in nonsingleton abstract isomorphism classes and links.)

Circumradii

To get from this to the squared circumradii, the results in the paper cited in Igor Pak's answer state that the squared circumradii of two abstractly isomorphic polyhedra should be roots of the same polynomial (actually we require some generalization of those theorems to non-simplicial polyhedra, which might be straightforward).

One thing to keep in mind is that the polynomial equations satisfied by general polynomial invariants (in the sense of Fedorchuk and Pak's paper) could be reducible. So their results alone are not sufficient to show that circumradii of uniform polyhedra that are abstractly isomorphic are actually Galois conjugate.

Indeed, the fact that the abstract isomorphism classes above don't all correspond to the Galois conjugacy classes found previously bear this out!

This is of course a good conceptual framework to think about what's going on, but a more hands-on approach is also possible:

Messer gives explicit expressions for the circumradii in his paper; in section 8, the circumradius ${}_{0}R$ is expressed as $\csc\phi$, where $\phi$ is the angle subtended by a half-edge when viewed from the center of the circumsphere. In section 9, he gives expressions for $\cot\phi$ for the uniform polyhedra for the first 3 of the 4 "forms" mentioned in Tito Pieza III's answer. Appendix A contains polynomials that define $Y=2-\tan^2\phi$ for the fourth "snub" case, and in fact the table given there places the Wythoff symbols together when their $Y$ satisfy the same polynomial. In each case, it should be possible to check the conjugacy from these expressions (after applying some trigonometric identities and rewriting certain cosines of rational multiples of $\pi$ as roots of polynomials).

actual abstract isomorphism classes
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j.c.
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Igor Pak's answer suggests that conjugate polyhedra in the sense of the question should be different realizations of the same abstract polyhedron (defined by a face lattice); so in this answer I'll try to figure outlist those which uniform polyhedra are abstractly isomorphic. The othertwo main referencereferences for this answer isare the paper "Closed-Form Expressions for Uniform Polyhedra and Their Duals" by Peter Messer and the page "Polytopes & their Incidence Matrices" by Richard Klitzing.

For uniform polyhedra, it suffices to consider the "vertex cycle symbol" which is a list of the $k$ faces around each vertex (up to cyclic permutations). Since there is just one kindIn earlier revisions of vertex, this information is enough to reconstruct the polyhedron and the face lattice; see Section 3 of Messer's paper for an explanation of the notation and howanswer I attempted to get themsay something about abstract isomorphism from the Wythoff symbols. Appendix B of Messer's paper provides a convenient listtable of the (non-prismatic) uniform polyhedra by their Wythoff symbols and vertex cycle symbols, see also this Wikipedia list, where these are listed in the "vertex figure" column.

Using these tablesMesser's paper, it's easybut as I wasn't able to check the following necessary condition for abstract isomorphism for the entries in Tito Pieza III's list: conjugate polyhedra have the same vertex cycle symbolsget very far with that approach, provided we only look at the numerators of the labels that appearI've deleted it. The denominator of a face in the vertex cycle symbol encodes how the face is realized; if the denominator is 1, then it's a convex polygon, but if it's greater than 1, it's a self-intersecting starfirst two comments to this answer refer to that text.)

For example, consider the first conjugate pairKlitzing's page seems to be unique in Tito Pieza III's listthat they indicate which have Wythoff symbols $5|2\, 3$ and $5/2|2\, 3$. The vertex cycle symbols are $3^5$ and $3^{5/2}$ respectively. Both of these symbols encode a cyclesets of 5 triangles around a vertex. In the former case, the five triangles are laid out in the usual way without intersection and "winding number 1" and in the latter case, they have winding 2. Thesepolyhedra are easily seen to be abstractly isomorphic, but it's unfortunately a bit tricky to navigate (as they are both realizationsI could not find an easy to parse list of just the abstract icosahedron).

As another more interesting example, consider75 non-prismatic uniform polytopes; the pair $3|3\, 5/2$ ($U_{30}$) and $3/2|3\, 5$closest was ($U_{47}$this). Their vertex cycle symbols are $(3\cdot5/2)^3$ and $(3\cdot5)^{3/2}$, respectively. In the first case we haveWith (triangle, pentagram, triangle, pentagram, triangleWikipedia's list of uniform polyhedra, pentagram) aroundI managed to find each vertex with winding 1 and inof the second case, we have (triangle, pentagon, triangle, pentagon, triangle, pentagon) around each vertex with winding 2. Both polyhedra haveon his site by googling their names with the same face latticekeyword "site:bendwavy. This pair is not in Tito Pieza's list, since the squared circumradii of both polyhedra are both $3/2$ which is degree 1, however there's still a sense in which they are conjugate!org".

Here's an example from the comments showing that this is not enough, consider the uniform polyhedra corresponding to the Wythoff symbols $\frac{5}{3}\, \frac{5}{2}|3$ ($U_{62}$), $\frac{5}{4}\, 5|3$ ($U_{65}$), and $\frac{5}{3}\, 5|3$ ($U_{44}$). The vertex cycle symbols are $(\frac{5}{2}\cdot6\cdot\frac{5}{3}\cdot6)$, $(5\cdot6\cdot\frac{5}{4}\cdot6)$ and $(5\cdot6\cdot\frac{5}{3}\cdot6)$ which all have the same numerators. While the first two are abstractly isomorphic (as we can see from these pages), the third polyhedron has twice the number of faces, edges and vertices of the first two!a summary:

The conjugate groupings of uniform polyhedra in Tito Pieza III's list correspond to abstract isomorphism classes of uniform polyhedra except that:

  • 13 and 14 in his list form one abstract isomorphism class ($U_{33},U_{61},U_{43},U_{42}$ are all abstractly isomorphic),
  • 23 ($U_{46}$) is actually abstractly isomorphic to $U_{64}$,
  • $U_{30}$ and $U_{47}$ are abstractly isomorphic, but not on the list of conjugates,
  • $U_{62}$ and $U_{65}$ are also abstractly isomorphic and not on the list of conjugates,
  • and finally, 22 ($U_{12}$) is in an abstract isomorphism class on its own, as are $U_n$ with $n=1, 2,3,4,5,6,7,8,15,16,36,38,41,44,45,56,59,62,65,75$ (the same list as in his comment on that answer, except that 30, 47, 62, 64, 65 are omitted, per above).

I'm not sure if there's a quick way to determine from the vertex cycle symbol whether two uniform polyhedra are abstractly isomorphic or not. However, it shouldn't take too long to go through(Here's the text document I compiled while searching Klitzing's site that includes the links to eachnames of the polyhedra on this page of Richard Klitzing's to figure out which polyhedra are abstractly isomorphicin nonsingleton abstract isomorphism classes and links.)

To get from this to the squared circumradii, the results in the paper cited in Igor Pak's answer state that the squared circumradii of two conjugateabstractly isomorphic polyhedra should be roots of the same polynomial (actually we require some generalization of those theorems to non-simplicial polyhedra, which might be straightforward).

Added: one thing to keep in mind is that the polynomial equations satisfied by general polynomial invariants (in the sense of Fedorchuk and Pak's paper) could be reducible. So their results alone are not sufficient to show that circumradii of uniform polyhedra that are abstractly isomorphic are actually Galois conjugate.

One thing to keep in mind is that the polynomial equations satisfied by general polynomial invariants (in the sense of Fedorchuk and Pak's paper) could be reducible. So their results alone are not sufficient to show that circumradii of uniform polyhedra that are abstractly isomorphic are actually Galois conjugate.

Indeed, the fact that the abstract isomorphism classes above don't all correspond to the Galois conjugacy classes found previously bear this out!

Igor Pak's answer suggests that conjugate polyhedra in the sense of the question should be different realizations of the same abstract polyhedron (defined by a face lattice); so in this answer I'll try to figure out which uniform polyhedra are abstractly isomorphic. The other main reference for this answer is the paper "Closed-Form Expressions for Uniform Polyhedra and Their Duals" by Peter Messer.

For uniform polyhedra, it suffices to consider the "vertex cycle symbol" which is a list of the $k$ faces around each vertex (up to cyclic permutations). Since there is just one kind of vertex, this information is enough to reconstruct the polyhedron and the face lattice; see Section 3 of Messer's paper for an explanation of the notation and how to get them from the Wythoff symbols. Appendix B of Messer's paper provides a convenient list of the (non-prismatic) uniform polyhedra by their Wythoff symbols and vertex cycle symbols, see also this Wikipedia list, where these are listed in the "vertex figure" column.

Using these tables, it's easy to check the following necessary condition for abstract isomorphism for the entries in Tito Pieza III's list: conjugate polyhedra have the same vertex cycle symbols, provided we only look at the numerators of the labels that appear. The denominator of a face in the vertex cycle symbol encodes how the face is realized; if the denominator is 1, then it's a convex polygon, but if it's greater than 1, it's a self-intersecting star.

For example, consider the first conjugate pair in Tito Pieza III's list which have Wythoff symbols $5|2\, 3$ and $5/2|2\, 3$. The vertex cycle symbols are $3^5$ and $3^{5/2}$ respectively. Both of these symbols encode a cycle of 5 triangles around a vertex. In the former case, the five triangles are laid out in the usual way without intersection and "winding number 1" and in the latter case, they have winding 2. These are easily seen to be abstractly isomorphic (as they are both realizations of the abstract icosahedron).

As another more interesting example, consider the pair $3|3\, 5/2$ ($U_{30}$) and $3/2|3\, 5$ ($U_{47}$). Their vertex cycle symbols are $(3\cdot5/2)^3$ and $(3\cdot5)^{3/2}$, respectively. In the first case we have (triangle, pentagram, triangle, pentagram, triangle, pentagram) around each vertex with winding 1 and in the second case, we have (triangle, pentagon, triangle, pentagon, triangle, pentagon) around each vertex with winding 2. Both polyhedra have the same face lattice. This pair is not in Tito Pieza's list, since the squared circumradii of both polyhedra are both $3/2$ which is degree 1, however there's still a sense in which they are conjugate!

Here's an example from the comments showing that this is not enough, consider the uniform polyhedra corresponding to the Wythoff symbols $\frac{5}{3}\, \frac{5}{2}|3$ ($U_{62}$), $\frac{5}{4}\, 5|3$ ($U_{65}$), and $\frac{5}{3}\, 5|3$ ($U_{44}$). The vertex cycle symbols are $(\frac{5}{2}\cdot6\cdot\frac{5}{3}\cdot6)$, $(5\cdot6\cdot\frac{5}{4}\cdot6)$ and $(5\cdot6\cdot\frac{5}{3}\cdot6)$ which all have the same numerators. While the first two are abstractly isomorphic (as we can see from these pages), the third polyhedron has twice the number of faces, edges and vertices of the first two!

I'm not sure if there's a quick way to determine from the vertex cycle symbol whether two uniform polyhedra are abstractly isomorphic or not. However, it shouldn't take too long to go through the links to each of the polyhedra on this page of Richard Klitzing's to figure out which polyhedra are abstractly isomorphic.

To get from this to the squared circumradii, the results in the paper cited in Igor Pak's answer state that the squared circumradii of two conjugate polyhedra should be roots of the same polynomial (actually we require some generalization of those theorems to non-simplicial polyhedra, which might be straightforward).

Added: one thing to keep in mind is that the polynomial equations satisfied by general polynomial invariants (in the sense of Fedorchuk and Pak's paper) could be reducible. So their results alone are not sufficient to show that circumradii of uniform polyhedra that are abstractly isomorphic are actually Galois conjugate.

Igor Pak's answer suggests that conjugate polyhedra in the sense of the question should be different realizations of the same abstract polyhedron (defined by a face lattice); so in this answer I'll list those which are abstractly isomorphic. The two main references for this answer are the paper "Closed-Form Expressions for Uniform Polyhedra and Their Duals" by Peter Messer and the page "Polytopes & their Incidence Matrices" by Richard Klitzing.

(In earlier revisions of this answer I attempted to say something about abstract isomorphism from the table of vertex cycle symbols in Messer's paper, but as I wasn't able to get very far with that approach, I've deleted it. The first two comments to this answer refer to that text.)

Klitzing's page seems to be unique in that they indicate which sets of polyhedra are abstractly isomorphic, but it's unfortunately a bit tricky to navigate (I could not find an easy to parse list of just the 75 non-prismatic uniform polytopes; the closest was this). With Wikipedia's list of uniform polyhedra, I managed to find each of the polyhedra on his site by googling their names with the keyword "site:bendwavy.org".

Here's a summary:

The conjugate groupings of uniform polyhedra in Tito Pieza III's list correspond to abstract isomorphism classes of uniform polyhedra except that:

  • 13 and 14 in his list form one abstract isomorphism class ($U_{33},U_{61},U_{43},U_{42}$ are all abstractly isomorphic),
  • 23 ($U_{46}$) is actually abstractly isomorphic to $U_{64}$,
  • $U_{30}$ and $U_{47}$ are abstractly isomorphic, but not on the list of conjugates,
  • $U_{62}$ and $U_{65}$ are also abstractly isomorphic and not on the list of conjugates,
  • and finally, 22 ($U_{12}$) is in an abstract isomorphism class on its own, as are $U_n$ with $n=1, 2,3,4,5,6,7,8,15,16,36,38,41,44,45,56,59,62,65,75$ (the same list as in his comment on that answer, except that 30, 47, 62, 64, 65 are omitted, per above).

(Here's the text document I compiled while searching Klitzing's site that includes the names of the polyhedra in nonsingleton abstract isomorphism classes and links.)

To get from this to the squared circumradii, the results in the paper cited in Igor Pak's answer state that the squared circumradii of two abstractly isomorphic polyhedra should be roots of the same polynomial (actually we require some generalization of those theorems to non-simplicial polyhedra, which might be straightforward).

One thing to keep in mind is that the polynomial equations satisfied by general polynomial invariants (in the sense of Fedorchuk and Pak's paper) could be reducible. So their results alone are not sufficient to show that circumradii of uniform polyhedra that are abstractly isomorphic are actually Galois conjugate.

Indeed, the fact that the abstract isomorphism classes above don't all correspond to the Galois conjugacy classes found previously bear this out!

some corrections
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j.c.
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j.c.
j.c.
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