diff --git a/ufl/cell.py b/ufl/cell.py
index 9ef8e9975..086d57195 100644
--- a/ufl/cell.py
+++ b/ufl/cell.py
@@ -96,7 +96,7 @@ def num_facets(self) -> int:
 
     def num_ridges(self) -> int:
         """Get the number of ridges.
-
+        
         Ridges are entities of dimension tdim-2.
         """
         tdim = self.topological_dimension()
@@ -196,6 +196,10 @@ def peak_types(self) -> typing.Tuple[AbstractCell, ...]:
               ("triangle", "quadrilateral", "quadrilateral", "quadrilateral", "triangle"), ("prism", )],
     "pyramid": [tuple("vertex" for i in range(5)), tuple("interval" for i in range(8)),
                 ("quadrilateral", "triangle", "triangle", "triangle", "triangle"), ("pyramid", )],
+    "pentatope": [tuple("vertex" for i in range(5)), tuple("interval" for i in range(10)),
+                  tuple("triangle" for i in range(10)), tuple("tetrahedron" for i in range(5), ("pentatope", ))],
+    "tesseract": [tuple("vertex" for i in range(16)), tuple("interval" for i in range(32)),
+                  tuple("quadrilateral" for i in range(24)), tuple("hexahedron" for i in range(8)), ("tesseract", )],
 }
 
 
@@ -422,6 +426,8 @@ def simplex(topological_dimension: int, geometric_dimension: typing.Optional[int
         return Cell("triangle", geometric_dimension)
     if topological_dimension == 3:
         return Cell("tetrahedron", geometric_dimension)
+    if topological_dimension == 4:
+        return Cell("pentatope", geometric_dimension)
     raise ValueError(f"Unsupported topological dimension for simplex: {topological_dimension}")
 
 
@@ -435,6 +441,8 @@ def hypercube(topological_dimension, geometric_dimension=None):
         return Cell("quadrilateral", geometric_dimension)
     if topological_dimension == 3:
         return Cell("hexahedron", geometric_dimension)
+    if topological_dimension == 4:
+        return Cell("tesseract", geometric_dimension)
     raise ValueError(f"Unsupported topological dimension for hypercube: {topological_dimension}")
 
 
diff --git a/ufl/finiteelement/elementlist.py b/ufl/finiteelement/elementlist.py
index da3855641..6ea6a6fa2 100644
--- a/ufl/finiteelement/elementlist.py
+++ b/ufl/finiteelement/elementlist.py
@@ -12,6 +12,7 @@
 # Modified by Marie E. Rognes <meg@simula.no>, 2010
 # Modified by Lizao Li <lzlarryli@gmail.com>, 2015, 2016
 # Modified by Massimiliano Leoni, 2016
+# Modified by Robert Kloefkorn, 2022
 
 import warnings
 from numpy import asarray
@@ -82,12 +83,12 @@ def show_elements():
 #       the future, add mapping name as another element property.
 
 # Cell groups
-simplices = ("interval", "triangle", "tetrahedron")
-cubes = ("interval", "quadrilateral", "hexahedron")
+simplices = ("interval", "triangle", "tetrahedron", "pentatope")
+cubes = ("interval", "quadrilateral", "hexahedron", "tesseract")
 any_cell = (None,
             "vertex", "interval",
             "triangle", "tetrahedron", "prism",
-            "pyramid", "quadrilateral", "hexahedron")
+            "pyramid", "quadrilateral", "hexahedron", "pentatope", "tesseract")
 
 # Elements in the periodic table # TODO: Register these as aliases of
 # periodic table element description instead of the other way around