Cultural Background
Like many Native American designs, the bead loom is based on four-fold symmetry. The traditional bead loom existed before European contact, and new versions are still in use today.
Here are four images of Native American designs. How would you describe them in terms of geometry? What geometric features do they all have in common?
Pronunciation of tribes:
Shoshone: “show-SHOW-nee”
Pawnee: “PAW-nee”
Navajo: “NAV-uh-hoe”
Many animals have bodies in which the right half is the mirror image of the left half. That is called "reflection symmetry." Some things have reflection symmetry between top and bottom. Footballs and soup cans, for example, have one shape that is the same for top and bottom, and a different shape that is the same for left and right. If you have the same shape reflected in both directions, it is called "four-fold symmetry."
If we ignore some details, like the color differences in the bead work example, then all of these designs can be described as having four-fold symmetry. Four-fold symmetry is a deep design theme in many Native American cultures. It is used as an organizing principle for religion, society, and native technology. Many native languages, for example, use base four counting. Teepees are often made with four base poles, each placed in one of the four directions. Prayers are often offered to "the four winds."
Here are four images of Native American designs. How would you describe them in terms of geometry? What geometric features do they all have in common?
Pronunciation of tribes:
Shoshone: “show-SHOW-nee”
Pawnee: “PAW-nee”
Navajo: “NAV-uh-hoe”
Many animals have bodies in which the right half is the mirror image of the left half. That is called "reflection symmetry." Some things have reflection symmetry between top and bottom. Footballs and soup cans, for example, have one shape that is the same for top and bottom, and a different shape that is the same for left and right. If you have the same shape reflected in both directions, it is called "four-fold symmetry."
If we ignore some details, like the color differences in the bead work example, then all of these designs can be described as having four-fold symmetry. Four-fold symmetry is a deep design theme in many Native American cultures. It is used as an organizing principle for religion, society, and native technology. Many native languages, for example, use base four counting. Teepees are often made with four base poles, each placed in one of the four directions. Prayers are often offered to "the four winds."
Four Fold Symmetry
Navajo sand painting offers many good examples of four-fold symmetry. These are used in healing ceremonies. A medicine man ("hataalii") completes the drawing in one day, using colored powder such as crushed stone. The painting is brushed away later that night, along with the illness.
In the Navajo religion, the hataalii heals through the balance of forces. Sand paintings often use reflection symmetry to show these paired forces. Some of these structures are similar to a Cartesian graph. Navajo tradition does not permit photos of sand paintings, so at left we have shown a Navajo rug based on a sand painting. Here we see four human figures. Figures on the horizontal axis are hunched over with back packs. Those on the vertical axis have straight backs. As a Cartesian graph:
+X = Right-side up figure with backpack
-X = Upside-down figure with backpack
+Y = White straight figure
-Y = Dark straight figure
In the Navajo religion, the hataalii heals through the balance of forces. Sand paintings often use reflection symmetry to show these paired forces. Some of these structures are similar to a Cartesian graph. Navajo tradition does not permit photos of sand paintings, so at left we have shown a Navajo rug based on a sand painting. Here we see four human figures. Figures on the horizontal axis are hunched over with back packs. Those on the vertical axis have straight backs. As a Cartesian graph:
+X = Right-side up figure with backpack
-X = Upside-down figure with backpack
+Y = White straight figure
-Y = Dark straight figure
The Math/Science bead connection
Bead work can also help us think about math and science. In this image at top we can see how the atomic structure of diamond looks in a Scanning Electron Microscope: the carbon atoms are lined up in a layer, like bead work with holes in it. Of course this is just the surface, there is another layer below that, another below that, and so on.
However the carbon atoms in diamond are not just linked like beads in a single layer, they are also strongly linked between layers (lower left). Compare this to a diagram for graphite (lower right). Graphite is also made only of carbon atoms, but it is very soft because the layers of atoms are not strongly connected. That is why we use graphite for pencils: the layers slide off and stick to the paper.
However the carbon atoms in diamond are not just linked like beads in a single layer, they are also strongly linked between layers (lower left). Compare this to a diagram for graphite (lower right). Graphite is also made only of carbon atoms, but it is very soft because the layers of atoms are not strongly connected. That is why we use graphite for pencils: the layers slide off and stick to the paper.
Virtual Bead Loom
Like many other Native American knowledge systems, the bead loom is much like a Cartesian coordinate system. We have beads in rows and columns: which is the X axis and which is the Y axis? If Europeans had never come to North America, what names would we use for "X axis" and "Y axis"?
The Virtual Bead Loom simulates the same grid pattern as the traditional bead loom. Users place colored circles in columns (the Y-axis) and rows (the X-axis).
There are several tools for placing beads on the virtual loom. This is located under the goal images picture. In each case you use the "tab" key or the mouse to move your cursor to the field for entering the coordinates, then you enter them, and then press the button for the shape tool. The point tool places a single bead:
The line tool places lines of beads. You specify the two endpoints of the line. Diagonal lines tend to be jagged, but resizing the grid can help that.
The rectangle tool fills in a rectangle of beads. You specify two vertices (lower right and upper left). The rectangles of this tool are always aligned with the axes.
The triangle tool fills in a triangle of beads. You specify the three vertices.
The iterative triangle tool: Our first triangle tool made jagged edges, while traditional bead work has beautifully regular edges. We interviewed some native bead workers, and found that their algorithms were iterative. The triangle iteration tool reflects this tradition of indigenous mathematics. For example, the triangle in the bead work at the top of this page was made by adding one bead on each side of the row, every three rows, as you go in the -Y direction.
If you click on the pattern below, you will have a variety of pattern choices to chose from. Enjoy!
The Virtual Bead Loom simulates the same grid pattern as the traditional bead loom. Users place colored circles in columns (the Y-axis) and rows (the X-axis).
There are several tools for placing beads on the virtual loom. This is located under the goal images picture. In each case you use the "tab" key or the mouse to move your cursor to the field for entering the coordinates, then you enter them, and then press the button for the shape tool. The point tool places a single bead:
The line tool places lines of beads. You specify the two endpoints of the line. Diagonal lines tend to be jagged, but resizing the grid can help that.
The rectangle tool fills in a rectangle of beads. You specify two vertices (lower right and upper left). The rectangles of this tool are always aligned with the axes.
The triangle tool fills in a triangle of beads. You specify the three vertices.
The iterative triangle tool: Our first triangle tool made jagged edges, while traditional bead work has beautifully regular edges. We interviewed some native bead workers, and found that their algorithms were iterative. The triangle iteration tool reflects this tradition of indigenous mathematics. For example, the triangle in the bead work at the top of this page was made by adding one bead on each side of the row, every three rows, as you go in the -Y direction.
- "Direction" -- determines in which direction your rows will accumulate
- Starting at X,Y -- that is the center of the starting row.
- "After every ___ rows" -- lets you determine how many rows you go through before adding more beads to the end.
- "Add ___ to both ends" -- the number of beads that will be added on each side of the center each time.
- "For ___ rows in total" -- how many rows you will bead in this triangle.
If you click on the pattern below, you will have a variety of pattern choices to chose from. Enjoy!