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Moorish Rose - A geometric supplemental guide
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Moorish Rose - A geometric supplemental guide
Article © MAIL User: RheingoldRiver
This article is intended to be a companion to MaxumX's tutorial for Moorish Rose (located here: Moorish Rose (CGI)), which is what I used to learn the weave in the first place. The goal is to give some context for what you're doing in the tutorial, along with some explanation of the weave as a whole. Hopefully this will help you figure out potential mistakes and resolve them, and understand the weave a bit better before attempting it.
The first row is relatively easy to create. Just don't accidentally flip any rings over after adding them to your Mobius unit and you'll be fine. So this tutorial will start with the second row.
To see higher-resolution copies of the images included in this article, go here.
Figure 1 - A Hexagon Grid.
Moorish Rose is a hexagon grid formed by interlocking Mobius units. As in a hexagon grid, there are hexagons, edges between 2 hexagons, and vertices between 3 hexagons. The grid is divided into rows. Two rows here are distinguished, one in red and one in blue. The border between the rows is a zig zag, and there are upper vertices (orange) and lower vertices (green).
If we color the next vertex in the second row blue, we will finish the orange "upper" vertex and also the "lower" vertex after that. In fact, once one row is complete, every hexagon that we add (other than the first in a row) will complete two new vertices, one upper and one lower. (Does that make sense? Each hexagon has 6 vertices and each vertex has 3 hexagons, so there must be a 2:1 ratio of vertices to hexagons, so yes, adding 1 more hexagon should add 2 more vertices.)
Figure 2 - A Hexagon Grid superimposed on Moorish Rose.
Here we can really see that a hexagon grid fits well onto the weave. The empty center of a Mobius corresponds to one hexagon, and each of the 6 rings forming the Mobius unit corresponds to one corner of the hexagon. What MaxumX calls a "flat" is really an edge of a hexagon.
Figure 3 - Terminology.
Since the first row is easy to make, we're assuming you've finished it and calling it the "old" row. You're now making the "new/current" row (row 2). Here are some other definitions that we'll use:
Vertex - The joining of 3 units at one point. As we'll see later, there are 9 rings that contribute to each vertex. Visualizing all 9 rings is difficult, but 3 of them are easy to see since they form a triangle.
Triangle - the center of a vertex, as circled in the diagram in red. These are important because they're a really good visual check that you're doing the right thing, so be familiar with them and their placement.
Unit - The six rings that make up a Mobius unit. There's an empty hole in the center of each unit. If you're thinking about the hex grid, a unit is a single hexagon.
Face - same as unit. "Unit," "Mobius," and "Face" will be used interchangeably.
Edge - aka a "Flat" in MaxumX's tutorial, the set of four rings belonging to neighboring units that mutually intersect each other.
If you've studied graph theory, I'm using "edge," "vertex," and "face" in their graph-theoretic definitions.
Figure 4 - Naming the grid.
It's a lot easier to describe what's going on if we assign some names to various features in the grid. I'll use the same names for the rest of the article.
The rows are 1, 2, and 3. 1 is the old row, 2 is the current row, and 3 is only included to make the picture easier to understand.
Each unit has been given a letter A through L. Mostly we will be discussing units A, B, C, E, and F.
Each vertex has a letter with a subscript: U1 through U3 are the lower vertices, and V1 through V3 are the upper vertices.
Finally, we will name edges based on what faces they're incident to (touching): AE is the edge between face A and E, etc.
Now we're going to get into really understanding the structure of the weave.
Figure 5 - Anatomy of an edge.
Here the green line delineates 3 edges: AB, EF, and IJ. We only care about EF right now. The orange and yellow rings contribute to EF from unit E, and the blue and purple rings contribute to it from unit F. Note that the yellow and purple rings each form one part of a triangle along the green line, and also note that the orange and blue rings do *not* do the same thing (they are parts of other triangles, but not ones along the green line).
The blue and purple both intersect both of orange and yellow (and vice versa), and since each pair is part of a unit (and so intersect each other), these 4 rings all intersect mutually. Furthermore, there are no other rings belonging to E going through F or to F going through E. This set of 4 cross-unit intersections uniquely defines the EF edge. Each time you create an edge, look to be adding two and only two rings of the new Mobius through each of two rings from the adjacent one.
Figure 6 - Anatomy of a vertex.
In this figure we look at the anatomy of a vertex, in this case we're focusing on vertex U1.
The red rings "hug" the triangle. Each belongs to one parent unit (A, B, or E), and each goes through BOTH of the other two faces incident to the vertex.
The pink rings form the triangle. Each goes through exactly ONE other face in addition to its parent (through the face counter-clockwise of its parent). B's pink goes through E; E's pink through A; and A's pink through B.
The green rings intersect through the face immediately CLOCKWISE of its parent: E's through B, B's through A, and A's through E.
As is the case with most characteristics of this weave, there is a simple and beautiful symmetry at play. These 9 rings are the only rings in the entire weave that go through any pair of the units A, B, and E.
One thing to note - if a face is 6 mutually intersecting rings, and an edge is 4 mutually intersecting rings, shouldn't the vertex only be the 3 red mutually intersecting rings? Yeah - I would probably call those 3 the vertex. But that doesn't provide as much insight to the weave as describing all 9 rings does, and also the inner triangles are by far the easiest part of the vertex to see, so we'll use this definition.
Figure 7 - Anatomy of a vertex (alternate coloring).
Here is an alternate coloring in case you prefer to look at this one. Vertex E's rings are in red and orange; B's are in yellow and green; and A's are in blue and purple. Red, yellow, and purple rings go through exactly ONE other Mobius involved in the vertex, while orange, yellow, and blue go through BOTH other Mobiuses.
The above discussion proves something useful to know about the weave: Each ring can be uniquely described by 2 things:
Its parent unit
The vertex of which it goes through both other units.
Now that we have described most of the geometry of the weave, let's get into the specifics of actually weaving it.
Figure 8 - The path of a ring.
The ring in question is highlighted in white (it's the F-V1 ring). Let's look at the rings it connects through. These are colored in rainbow order of how it goes through them, i.e. if you shrunk all of the intersecting rings so that they fit snugly around the white ring, in what order would lie? Or, put another way, in what order does the white ring intersect the planes defined by the colored rings?
However you want to put it, it's really important to understand this diagram, because every single ring has the exact same same connectivity path (this weave is awesomely symmetric!)
First, let's notice what structures this ring is part of:
Face F
Edge BF
Edge EF
Vertex V1 (or vertex EFB) (as a red twice-intersecting ring)
Vertex U2 (as a triangle (pink), once-intersecting ring)
Vertex EFI (unnamed in our diagram) (as a green once-intersecting ring)
One important point to note about connectivity is that a ring's connections with other units always alternates which unit it's going through, but its path never alternates between parent and adjacent.
So here is the path:
Adjacent unit (from parent B) - Red
Adjacent unit (from parent E) - Orange
Adjacent unit (from parent B) - Yellow
Adjacent unit (rom parent E) - Green
Parent unit - Blue
Parent unit - Purple
Parent unit - Pink
Parent unit - Grey
Parent unit - Brown
In my experience, the connectivity order is the easiest thing you will mess up when you try this wave. A ring is standing up? You probably went Red - Yellow - Orange instead of Red - Orange - Yellow. The order of the parent unit rings is unlikely to be messed up and is mostly listed here for completeness.
One final note before we get to actually weaving!
Prior to participating in an edge, the order of the rings in a Mobius is not defined. This is really important!!! Thinking that order matters before connections have been made might cause you to troubleshoot your weave incorrectly.
If you're trying to weave a ring through the "top" or the "bottom" of another unit, but that other unit isn't fully connected yet, then there are rings which are fully interchangeable. So don't worry about which is top. The one that's top is your favorite one. Or your least favorite one. It doesn't matter. Once connections have been made, the rings become distinguishable, and connecting through the proper one is important, but 2 rings that belong to the same parent and have no connections beyond those of their parent are indistinguishable.
And one non-structural piece of advice - after completing the first row, this weave is a lot easier to build if you lay it flat on a table or desk, even if you're used to weaving in the air. Once you're very familiar with the connectivity patterns, you can weave it in the air, but the first time you make it, keep it flat the whole time so nothing gets jostled out of position.
Ok! Time to actually weave now! Let's go through MaxumX's tutorial and identify what the rings are doing in each step. Remember to refer back to figure 4 for nomenclature. Again, creating the first row is relatively straightforward so we'll start with step 9.
In the upcoming descriptions, there are 3 "hard" steps where you have to pay attention to neighboring unit intersecting order. These steps are 10, 15, and 17. For every other step, you just need to choose the right rings to go through.
Steps 9-14: Create unit E.
Step 9: Finish the triangle U1, and create the first ring of the AE edge. This is the only triangle we'll finish with this unit since it's the start of a new row.
Step 10: Finish the AE edge, and start the BE edge. This is the first "hard" ring, so remember the ring path diagram for this one. In this case you're going through B outer then A inner then B inner then A outer.
Step 11: Finish the BE edge, and now we're done adding rings that will intersect with pre-existing units for unit E.
Steps 12-14: Finish the E unit by just adding in extra Mobius rings. Yay! Your first second-row unit is done!
Steps 15-20: Create unit F. If you can do this, you can do the entire weave!
Step 15: Finish triangle U2, and make the first ring for edges BF and EF. Remember intersection order! (This ring is the same as the one we added in step 10, just rotated a bit.) (haha that was sort of a joke because all rings are the same, but seriously this one has the same set of pre-existing rings that it has to travel through as the one in step 10)
Step 16: Finish edge EF, and create triangle V1. You can actually reverse steps 16 and 17 if you like, since that way the "hard" ring of 17 has to pass through one less unit overall; I prefer to do it 17 then 16 but either works.
Step 17: Finish edge BF, and create the first ring for edge CF. Again be careful about the intersection order. This is the last hard ring!
Step 18: Finish edge CF. Congrats! If you got this far and everything is right, you've basically mastered the entire weave!
Steps 19-20: Finish that Mobius unit.
Depending on the shape of your overall pattern, when you get to unit H, you might have to modify the instructions slightly since there's no unit to the right of D. So for that unit, still follow steps 15-20, but in steps 17 and 18 there's no bottom-right rings to connect through at all.
The first row is relatively easy to create. Just don't accidentally flip any rings over after adding them to your Mobius unit and you'll be fine. So this tutorial will start with the second row.
To see higher-resolution copies of the images included in this article, go here.
Figure 1 - A Hexagon Grid.
Moorish Rose is a hexagon grid formed by interlocking Mobius units. As in a hexagon grid, there are hexagons, edges between 2 hexagons, and vertices between 3 hexagons. The grid is divided into rows. Two rows here are distinguished, one in red and one in blue. The border between the rows is a zig zag, and there are upper vertices (orange) and lower vertices (green).
If we color the next vertex in the second row blue, we will finish the orange "upper" vertex and also the "lower" vertex after that. In fact, once one row is complete, every hexagon that we add (other than the first in a row) will complete two new vertices, one upper and one lower. (Does that make sense? Each hexagon has 6 vertices and each vertex has 3 hexagons, so there must be a 2:1 ratio of vertices to hexagons, so yes, adding 1 more hexagon should add 2 more vertices.)
Figure 2 - A Hexagon Grid superimposed on Moorish Rose.
Here we can really see that a hexagon grid fits well onto the weave. The empty center of a Mobius corresponds to one hexagon, and each of the 6 rings forming the Mobius unit corresponds to one corner of the hexagon. What MaxumX calls a "flat" is really an edge of a hexagon.
Figure 3 - Terminology.
Since the first row is easy to make, we're assuming you've finished it and calling it the "old" row. You're now making the "new/current" row (row 2). Here are some other definitions that we'll use:
If you've studied graph theory, I'm using "edge," "vertex," and "face" in their graph-theoretic definitions.
Figure 4 - Naming the grid.
It's a lot easier to describe what's going on if we assign some names to various features in the grid. I'll use the same names for the rest of the article.
Now we're going to get into really understanding the structure of the weave.
Figure 5 - Anatomy of an edge.
Here the green line delineates 3 edges: AB, EF, and IJ. We only care about EF right now. The orange and yellow rings contribute to EF from unit E, and the blue and purple rings contribute to it from unit F. Note that the yellow and purple rings each form one part of a triangle along the green line, and also note that the orange and blue rings do *not* do the same thing (they are parts of other triangles, but not ones along the green line).
The blue and purple both intersect both of orange and yellow (and vice versa), and since each pair is part of a unit (and so intersect each other), these 4 rings all intersect mutually. Furthermore, there are no other rings belonging to E going through F or to F going through E. This set of 4 cross-unit intersections uniquely defines the EF edge. Each time you create an edge, look to be adding two and only two rings of the new Mobius through each of two rings from the adjacent one.
Figure 6 - Anatomy of a vertex.
In this figure we look at the anatomy of a vertex, in this case we're focusing on vertex U1.
As is the case with most characteristics of this weave, there is a simple and beautiful symmetry at play. These 9 rings are the only rings in the entire weave that go through any pair of the units A, B, and E.
One thing to note - if a face is 6 mutually intersecting rings, and an edge is 4 mutually intersecting rings, shouldn't the vertex only be the 3 red mutually intersecting rings? Yeah - I would probably call those 3 the vertex. But that doesn't provide as much insight to the weave as describing all 9 rings does, and also the inner triangles are by far the easiest part of the vertex to see, so we'll use this definition.
Figure 7 - Anatomy of a vertex (alternate coloring).
Here is an alternate coloring in case you prefer to look at this one. Vertex E's rings are in red and orange; B's are in yellow and green; and A's are in blue and purple. Red, yellow, and purple rings go through exactly ONE other Mobius involved in the vertex, while orange, yellow, and blue go through BOTH other Mobiuses.
The above discussion proves something useful to know about the weave: Each ring can be uniquely described by 2 things:
Now that we have described most of the geometry of the weave, let's get into the specifics of actually weaving it.
Figure 8 - The path of a ring.
The ring in question is highlighted in white (it's the F-V1 ring). Let's look at the rings it connects through. These are colored in rainbow order of how it goes through them, i.e. if you shrunk all of the intersecting rings so that they fit snugly around the white ring, in what order would lie? Or, put another way, in what order does the white ring intersect the planes defined by the colored rings?
However you want to put it, it's really important to understand this diagram, because every single ring has the exact same same connectivity path (this weave is awesomely symmetric!)
First, let's notice what structures this ring is part of:
One important point to note about connectivity is that a ring's connections with other units always alternates which unit it's going through, but its path never alternates between parent and adjacent.
So here is the path:
In my experience, the connectivity order is the easiest thing you will mess up when you try this wave. A ring is standing up? You probably went Red - Yellow - Orange instead of Red - Orange - Yellow. The order of the parent unit rings is unlikely to be messed up and is mostly listed here for completeness.
One final note before we get to actually weaving!
Prior to participating in an edge, the order of the rings in a Mobius is not defined. This is really important!!! Thinking that order matters before connections have been made might cause you to troubleshoot your weave incorrectly.
If you're trying to weave a ring through the "top" or the "bottom" of another unit, but that other unit isn't fully connected yet, then there are rings which are fully interchangeable. So don't worry about which is top. The one that's top is your favorite one. Or your least favorite one. It doesn't matter. Once connections have been made, the rings become distinguishable, and connecting through the proper one is important, but 2 rings that belong to the same parent and have no connections beyond those of their parent are indistinguishable.
And one non-structural piece of advice - after completing the first row, this weave is a lot easier to build if you lay it flat on a table or desk, even if you're used to weaving in the air. Once you're very familiar with the connectivity patterns, you can weave it in the air, but the first time you make it, keep it flat the whole time so nothing gets jostled out of position.
Ok! Time to actually weave now! Let's go through MaxumX's tutorial and identify what the rings are doing in each step. Remember to refer back to figure 4 for nomenclature. Again, creating the first row is relatively straightforward so we'll start with step 9.
In the upcoming descriptions, there are 3 "hard" steps where you have to pay attention to neighboring unit intersecting order. These steps are 10, 15, and 17. For every other step, you just need to choose the right rings to go through.
Steps 9-14: Create unit E.
Step 9: Finish the triangle U1, and create the first ring of the AE edge. This is the only triangle we'll finish with this unit since it's the start of a new row.
Step 10: Finish the AE edge, and start the BE edge. This is the first "hard" ring, so remember the ring path diagram for this one. In this case you're going through B outer then A inner then B inner then A outer.
Step 11: Finish the BE edge, and now we're done adding rings that will intersect with pre-existing units for unit E.
Steps 12-14: Finish the E unit by just adding in extra Mobius rings. Yay! Your first second-row unit is done!
Steps 15-20: Create unit F. If you can do this, you can do the entire weave!
Step 15: Finish triangle U2, and make the first ring for edges BF and EF. Remember intersection order! (This ring is the same as the one we added in step 10, just rotated a bit.) (haha that was sort of a joke because all rings are the same, but seriously this one has the same set of pre-existing rings that it has to travel through as the one in step 10)
Step 16: Finish edge EF, and create triangle V1. You can actually reverse steps 16 and 17 if you like, since that way the "hard" ring of 17 has to pass through one less unit overall; I prefer to do it 17 then 16 but either works.
Step 17: Finish edge BF, and create the first ring for edge CF. Again be careful about the intersection order. This is the last hard ring!
Step 18: Finish edge CF. Congrats! If you got this far and everything is right, you've basically mastered the entire weave!
Steps 19-20: Finish that Mobius unit.
Depending on the shape of your overall pattern, when you get to unit H, you might have to modify the instructions slightly since there's no unit to the right of D. So for that unit, still follow steps 15-20, but in steps 17 and 18 there's no bottom-right rings to connect through at all.
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