The document discusses the Klein quartic surface, which is a type of quartic surface defined by a degree 4 polynomial in projective 3-space. It has some unique properties including order-3 and order-7 rotational symmetries due to its tiling by heptagons and triangles. The Klein quartic has 168 total symmetries and exists fundamentally as a 3-holed torus. While it cannot be fully realized in R3, it can be modeled in other ways that preserve some attributes like rotational symmetry while sacrificing others like its curved nature.

1. A quartic surface is defined by a polynomial of degree 4 in A3: 𝒇 𝒙, 𝒚, 𝒛 = 𝟎. In this case, we term it an
affine surface. Similarly, the projective incarnation (P3
R) is defined by the real-valued solutions to the
homogenous degree 4 polynomial within the projective space of 4 parameters: 𝒇 𝒙, 𝒚, 𝒛, 𝒘 = 𝟎.
By implementing modern tools in algebraic geometry, we can uncover how the Klein
quartic relates to all other quartics and devise methods that describe its characteristics.
These methods include, but are not limited to, the discovery and explicit definitions of
lines, curves, and singularities that exist within the Klein model, but also the formation
of supportive algebraic constructs. The behavior of the Klein quartic provides us with a
unique understanding of quartic curves and surfaces—as it exists fundamentally as
both—and sketches an intricate portrait of rotation and symmetry for these kinds of
objects. The existence of quartics such as the Klein surface encourages a renewed
fascination with abstract objects in higher dimensions as well as the inspiration to unite
these nebulous structures in new, enlightening ways.
P3 (projective space): P3 is projective space of dimension 3. This means that it is defined by 4
homogenous coordinates or variables (𝑤, 𝑥, 𝑦, 𝑧) such that all polynomials within its space are homogenous and of
degree 3. In essence, we have the analog real space R3 (𝑥, 𝑦, 𝑧) joined together with the projective parameter w at
infinity. P3 can be thought of as the space described above such that the set of all unique projective coordinates
abides by the equivalence relation: 𝑤0, 𝑥0, 𝑦0, 𝑧0 = 𝜆(𝑤0, 𝑥0, 𝑦0, 𝑧0) in P3. 0, 0, 0, 0 is not allowed in
projective space, as it violates the inherent assumption above about projective space at infinity. Thus, P3 has
dimension 3. We can also carve projective 3-space into two main projective subspaces: the so-called real P3
R (or
RP2) or complex P3
C (or CP2) projective spaces which, in addition, abide by the algebraic rules of their respective
topologies.
Algebraic variety: An algebraic variety 𝑉(𝑓1, 𝑓2, … , 𝑓𝑛) is the set of common zeroes of polynomials
𝑓1, 𝑓2, … , 𝑓𝑛. Similarly, we can think of an algebraic variety as the zero-locus of a set of polynomial functions
within a given algebraic space.
Irreducible variety: An irreducible variety is a variety that cannot be constructed from disjoint sets that
are closed under the Zariski topology, i.e. a variety that cannot be constructed from two proper algebraic subsets
[1].
Singularity: A singularity is a point on a variety where the tangent space is not well-defined. On an algebraic
surface, we may realize this as a point at which the tangent plane is not well-defined. Thus, in P3, we can define
such singular points on a surface as the set of points (𝑥0, 𝑥1, … , 𝑥 𝑛) at which, for a function 𝑓 𝑥0, 𝑥1, … , 𝑥3
defining the surface has partial derivatives vanishing, i.e. of 𝑓 are 0, i.e. (𝑓𝑥0
, 𝑓𝑥1
, … , 𝑓𝑥3
) = 𝟎.
Dimension: The dimension of a vector space is the number of vectors required to construct a basis for that
entire space. For example, in R3, we have dimension 3, one for each of the x-, y-, and z- axes. Within algebraic
geometry, since varieties are described as common zero-loci, P3 has dimension 3, as locally it is isomorphic to R3.
The dimension of a variety X is defined as a dimension of the tangent space at a non-singular point of X.
Affine Open Subsets
Closed sets under the Zariski topology in P3 are given by 𝑉𝑓 = 𝑥, 𝑦, 𝑧, 𝑤 : 𝑓 𝑥, 𝑦, 𝑧, 𝑤 = 0 , then we may
define open sets similarly as 𝑈𝑓 = 𝑷3 − 𝑉𝑓. We can map these subsets to the affine case, giving rise to our affine
open subsets 𝑈 𝑥, 𝑈 𝑦, 𝑈𝑧, and 𝑈 𝑤, which are all isomorphic to R3.
We know that R3 ≅ 𝑈 𝑤 = { 𝑥, 𝑦, 𝑧, 1 }. Therefore, we can visually represent our surface, the Klein quartic, in
affine space A3. The figure to the left below depicts the case when 𝑤 = 1.
Is this point singular ??
The 3 figures above to the right depict the various affine chunks of the quartic when 𝒙, 𝒚, 𝒛 = 𝟏,
respectively. Because of the interchangeability of the variables in the quartic, each affine chunk is a
rotated view along some 𝒙−, 𝒚−, or 𝒛 − axis.
Singular Points
The Klein quartic is defined by the function (in P3):
𝒇 𝒙, 𝒚, 𝒛, 𝒘 = 𝒙 𝟑
𝒚 + 𝒚 𝟑
𝒛 + 𝒙𝒛 𝟑
= 𝟎
We can set w = 1 and examine the function as is without losing any real perspective. Thus, we must take the partial
derivatives of 𝑓 with respect to its parameters in order to investigate singularities that may exist on the Klein
quartic. Differentiating yields
𝑓𝑥 = 3𝑥2 𝑦 + 𝑧3 = 0
𝑓𝑦 = 3𝑦2 𝑧 + 𝑥3 = 0
𝑓𝑧 = 3𝑥𝑧2 + 𝑦3 = 0
Since the original quartic function is irreducible and since the varieties are interchangeable, it is sound to assume
that solutions to this system are of the form (𝑘, 𝑘, 𝑘), where k is a real constant. However, since one term has
constant multiple 3, such a point will not satisfy such a system for 𝑘 ≠ 0. The only such point that solves the
system is (0, 0, 0, 1), which is precisely the point depicted as the meeting of the two chunks of the Klein
quartic represented above as 𝑈 𝑤. So, this singular point can be termed a cusp, which we now know arises from the
centers of the heptagons plating the surface.
For a more elegant (and albeit brief) discussion of the singularities of the Klein quartic, one may peruse Elkies’
number theoretical approach to describing the quartic in his section of The Eightfold Way: The Beauty of Klein’s
Quartic Curve [6], which is referenced in this paper on a number of occasions.
The Klein quartic is thought of as the solutions, in complex projective coordinates, of the polynomial function:
𝒇(𝒙, 𝒚, 𝒛) = 𝒙 𝟑
𝒚 + 𝒚 𝟑
𝒛 + 𝒙𝒛 𝟑
= 𝟎
This cannot be fully realized in R3, as mentioned to the left. However, this function can be considered in P3. The
remarkable symmetries exhibited by the Klein quartic resonate most fundamentally as a 3-holed torus .When the
malleable dough of 3-holed torus is stretched, the abundant potential for symmetry becomes apparent as does its
tetrahedral tendencies.
 order-3 symmetry in the tiling of the heptagons
 order-7 symmetry in the arrangement of triangles -- 7 triangles meet each time at a vertex
• Both contribute to the tetrahedral form and the topological properties of the quartic itself:
The maximization of rotational capacity exhibited by the Klein quartic is a remarkable testament to its intricacy,
which Greg Egan points out stands in stark contrast to the continuous symmetry of a sphere, for example: “If you
think about it, it makes sense that a surface of genus greater than 1 will only have discrete symmetries, because the
presence of the extra holes will jam any kind of continuous sliding motion.”
• The identity element 𝒆 | No Rotation
• Rotations by 1/7 and 6/7 -- turns on 8 different heptagons | 48 elements – order 7
• Rotations by 1/3 or 2/3 -- turns on any of 28 different triangles| 56 elements – order 3
• Rotations by 1/4 or 3/4 -- flops pairs of edges -- 21 different sets| 42 elements – order 4
• rotations by 1/2 on the edges| 21 elements -- order 2 [3].
48 + 56 + 42 + 21 + 1 = 168 totalsymmetries
[1] "Algebraic Variety." Wikipedia. Wikimedia Foundation, 1 Apr. 2013. Web. 1 Apr. 2013
[2] Baez, John. "Klein's Quartic Curve." Klein's Quartic Curve. N.p., 04 June 2010. Web. 1 Apr. 2013.
<http://math.ucr.edu/home/baez/klein.html>.
[3] Egan, Greg. "Klein's Quartic Equation." Klein's Quartic Equation. N.p., n.d. Web. 2 Aug. 2006.
<http://gregegan.customer.netspace.net.au/SCIENCE/KleinQuartic/KleinQuarticEq.html>.
[4] Harris, Joe. Algebraic Geometry: a first course. Springer, GTM 133, 1992.
[5] "Klein Quartic." Wikipedia. Wikimedia Foundation, 04 Feb. 2013. Web. 1 Apr. 2013.
[6] Levy, Silvio. The Eightfold Way: The Beauty of Klein's Quartic Curve. Cambridge [England: Cambridge UP,
1999. Web.
[7] Matsieva, Julia. The Klein Quartic. N.p., 14 Dec. 2010. Web. 1 Apr. 2013
[8] "Quartic Surface." Wikipedia. Wikimedia Foundation, 27 Mar. 2013. Web. 1 Apr. 2013.
[9] "Riemann Surface." Wikipedia. Wikimedia Foundation, 04 Feb. 2013. Web. 1 Apr. 2013.
[10] Stay, Mike. N.p., n.d. Web. 1 Apr. 2013. <http://math.ucr.edu/~mike/klein/>.
[11] Weisstein, Eric W. "Klein Quartic." From MathWorld--A Wolfram Web Resource. 1 April. 2013.
http://mathworld.wolfram.com/KleinQuartic.html
[12] Westendorp, Gerard. "Platonic Tesselations of Riemann Surfaces." Platonic Tesselations of Riemann Surfaces.
N.p., n.d. Web. 1 Apr. 2013. <http://westy31.home.xs4all.nl/Geometry/Geometry.html>.
I’d like to take this opportunity to thank our wonderful advisor and professor Dr.
Ivona Grzegorczyk as well as classmates in the CSUCI Mathematics
Department for providing such a positive, stimulating learning environment!
𝒇∗
= 𝒂𝒙 𝟑
𝒚 + 𝒚 𝟑
𝒛 + 𝒙𝒛 𝟑
= 𝟎, with assumed 𝒘 = 𝟏
As parameter 𝑎 decreases towards 0, the 𝑥𝑦 plane in A3 begins to smooth out, revealing less actual
projective space between the “subsets” of the quartic.
As 𝒂 approaches 𝟎 :
𝒂 = 𝟏 𝒂 = 𝟑
𝟒 𝒂 = 𝟏
𝟐 𝒂 = 𝟏
𝟒 𝒂 = 𝟑
𝟒 𝒂 = 𝟎
As 𝒂 approaches +∞ :
𝒂 = 𝟏 𝒂 = 𝟐 𝒂 = 𝟒 𝒂 = 𝟏𝟎 𝒂 = 𝟏𝟎𝟎 𝒂 = 𝟏𝟎 𝟓
Beauty of the Klein Quartic Surface
Dev Ananda Advisor: Dr. Ivona Grzegorczyk Mini Grant Research Project Math 584
The Klein quartic can be modeled in other ways that preserve its rotational symmetry
but forgo its curved nature in the interests of topological authenticity and its
fundamental existence as a 3-holed torus. As mentioned before, the Klein Surface
cannot be realized in three dimensions visually, but can be modeled by 3-dimensional
objects.
Representations of the Klein Quartic
that preserve certain attributes and concede others…
(from left-to-right: As a tetrahedron, as a truncated cube, as a heptagonal surface)
Acknowledgements
ABSTRACT
Modeling the Klein Quartic
DEFINITIONS
RESULTS
References
DEFORMING the Klein Surface…
SYMMETRIES