LEGO® techniques with reflected wedges: Escaping the grid

After the popularity of his first guest post for us, Arno Knobbe (@legoarno on Instagram) returns today to explain more core building techniques for sending your models off the rectilinear grid. 

In my previous post, I discussed a classic LEGO® technique for building at an angle, the rectangle trick, and its relation to the "sugar grid", a concept recently introduced on the Brick Sculpt YouTube channel. In this article, we’ll look into another mainstay of LEGO building techniques, reflected wedges. I’ll investigate some parallels between this technique and the sugar grid as well.

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Reflected wedges

At its core, a pair of reflected wedges is simply two mirrored, but otherwise identical wedges (plates or bricks), where the diagonal edges are placed against one another:

As you may have noticed, diagonal edges are typically not of integer length (worse, they’re often irrational), so it’s hard to find matching parts outside the wedge family. The beauty of the reflected wedges technique is that you take two mirrored parts, and their awkward sides line up perfectly (this works for some human relationships also).

The earlier rectangle trick and this reflected-wedges technique appear disjunct, but are really two sides of the same coin, as the following demonstrates:

The two wedges (triangles in this case) are just two matching halves of the two rectangles shown, and their interface is the diagonal that was used to align the white and red rectangle at an angle.

Reflected wedges are excellent for attaching entire subassemblies at an angle. You can repeat the technique to form bigger wedges, such as depicted in the image below:

© Deepak Shenoy, reproduced with kind permission

This image – from Deepak Shenoy’s excellent The LEGO Builder’s Handbook – represents the first floor of the elegant set 10297 Boutique Hotel. The entire left section (of the hotel, not just of this floor) is placed at a 36.9° angle by placing three wedges against three identical, but mirrored wedges.

So how are these plates usually attached to one another? 

The obvious way to do this is using one or more hinge plates (or bricks), either above or below the reflected wedges. You don’t need to use all three of them as I have above (nor do you have to use the colours of the Dutch flag) but when using them, make sure you place the hinge’s pivot right over a grid corner, as portrayed here.

So in this era of the sugar grid, are hinge plates still the most obvious choice? Not necessarily, because you can actually attach the angled plate using studs of the sugar grid – provided the plate is big enough to cover sufficiently many studs.

As you may remember, sugar grids come in different ratios, with this particular grid being ratio 1:2 (from each blue stud, you move 1 module across and 2 up to the next). In the same way, the two mirrored wedges are of ratio 1:2. 

So what’s a 1:2 ratio wedge?

It simply means that the diagonal edge runs 1 module across and 2 up, going from one corner to the other. The above image shows a number of 1:2 pieces, including a curved one. Although these elements have different shapes, they have the same ratio and are thus compatible: their diagonal edges have the same angle. (The blue and white studs indicate where the connection points for the grid lie).

Shown above are:

• Wedge Plate 2 x 2 Right 24307 (Left 24299)
• Slope Curved 3 x 2 with Stud Notch Left 80177 (Right 80178)
• Wedge Plate 2 x 4 27° Right 65426 (Left 65429)
• Wedge Plate 4 x 6 47407, 29172
• Wedge Plate 6 x 4 Right 48205 (Left 48208)

The dual grid

Before moving on to grids and wedges of other ratios, let’s have a closer look at the 1:2 sugar grid again.

The grid’s ratio can be easily recognised from the number of modules up and across between the blue studs: 1 and 2. But have a look at the two darker shaded studs. These are 1 module across and 3 modules up!

Wait, what? One grid has two ratios? Indeed, each sugar grid has a primary ratio (1:2 in this case) and a dual ratio (1:3 here). This implies that each sugar grid supports two angles (on top of the normal 0° grid).

This may sound a bit surprising, but you’ve actually already seen these two angles. Take the following 4x4 square supported by a 1:2 sugar grid as an example.

The angle between the blue and white plate is 53.1° (computed from 2 x arctan(1/2)), whereas the angle between the blue and red plate is 36.9° (= 2 x arctan(1/3)). Together, 53.1° + 36.9° = 90°.

As a consequence, a 1:2 grid also supports reflected wedges of ratio 1:3. This is Wedge Plate 6 x 3 Left and Right (54384 and 54383)

Although I perhaps make it look easy, placing these reflected wedges on the grid is not straightforward, since you need to position them carefully relative to the grid. This is where the white, offset studs come in. While they tend to be in the way by knocking against the wedges’ edges, they do form helpful markers for where the corners of the wedges should be placed.

In this image, I coloured two of the grid’s offset studs red to indicate where the corners should go. But after pointing out these key locations, these markers should be removed for the wedges to fit properly. 

 

Reflected wedges are ideal for cleanly interfacing between the angled grid and the base grid, without leaving any gaps. 

Here’s an example of an angled 6x6 plate with a clean interface in two directions. Does this construct look vaguely familiar to you? 

Indeed, it featured almost literally in set 10305 Lion Knights’ Castle (still with those medieval hinges, though, no sugar grid in sight). 

Building at 45 degrees

The above combination of a 1:2 and 1:3 pair of reflected triangles leading to a 90° corner suggests an interesting opportunity: perhaps pairing an individual 1:2 wedge with a 1:3 wedge will lead to a 45° degree? You’ll have to trust me on this (because the proof is well beyond my math-license for this article) but the fit is indeed legal and mathematically sound:

I’ve used the incredibly useful A-frame plate (15706) here, to attach the two wedges at the required 45°, since there exists no sugar grid for this angle.

You could also use the ‘knee brace’ piece (79846, introduced in the 2021 LEGO® NINJAGO EVO sets) for this, which produces slightly different alignments:

Although this 1:2/1:3 combo has a pleasing elegance, it bothers me that there is no way to have both ends meet. This is the result of both diagonal edges being irrational in a non-compatible way (multiples of √5 and √10, being a factor √2 apart). 

Canadian builder Joel Short (Instagram, Flickr) found an interesting workaround, though:

By alternating between 1:2 and 1:3 along the edge, the far ends do meet up. Sort of. Due to the tip of the white 1:2 wedge sticking out slightly into the red 1:2 wedge, the connection is ever so slightly illegal. Not that Joel or I care…

Other ratios

Let’s review a few other ratios. 1:4 and 1:6 wedge plates are common, so here’s how they work:

For any grid with ratio n:m, you can find its dual ratio using the following formula:

(m-n):(m+n)

For 1:2, this formula indeed works as expected: (2-1):(2+1) = 1:3. For ratios 1:4 and 1:6, this produces 3:5 and 5:7, respectively. Unfortunately, no LEGO wedge plates sport such unusual ratios yet (TLG, are you reading this?). 

But wedges are not the only angled elements. It just so happens that the good old cheese slope (54200) has a ratio of 3:5, such that, when flipped on its side, it can be used as a companion of the 1:4 wedge, as maths student Luca Hermann (Instagram) brought to my attention.

Here’s a slightly clumsy but mathematically sound demonstration of a tan plate above a 1:4 grid with interfaces to both sides of the base plate.

With the following two exotic pairs of wedges, we start to venture into decidedly geeky territory:

The unusual construct at the front relies on the 2:3 ratio of the Wedge 4 x 4 Pointed (22391) and its dual ratio 1:5 of the Wedge 2 x 16 Triple (30382). The pointed wedge is held in place with a dark grey A-frame at the bottom.

The winglike assembly in the back shows a combo of 2:5 and 3:7, for the grille center (30036) and two winglets on the side (3933 and 3934), respectively. Again, two A-frames tie everything together quite nicely.

MOC by Tom Loftus

From all the wedge combos featured in this article, the one with the black pointed wedge felt the hardest to me to incorporate in an actual MOC. But one of New E’s fine contributors, Tom Loftus, got inspired by exactly this combo and produced a futuristic ship for us:

READ MORE: Tom shows how he built the ship and shares more pictures

Conclusion

So much for our analysis of LEGO kites! In a follow-up, I will be discussing triangles, specifically triangles without any irrational sides (New E readers being very rational folks). I will show how Pythagorean triangles relate to the two core angling techniques presented so far (the rectangle trick and reflected wedges), all on a solid foundation of the sugar grid…

 

READ MORE: Amelia Ticket Booth GwP review

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