Bicycle wheels are mathematically symmetrical.

Bicycle wheels are mathematically symmetrical. An excellent example of perfection.

They are rated according a perfect mathematical ratio, of a number of front chain wheels to the number of rear wheel sprockets. Rear wheels have either 1, 5, 6, or 7 sprockets sometimes 8.

Single front chain wheels are mathematically, one 1, one 5, one 6, one 7 and one 8, giving one, five, six, seven and eight speed bikes. Just adding a second chain wheel changes things. In this case two 1's, two 5's, two 6's,  two 7's and two 8 speed bikes.

Adding a 3rd, results in three 1's ones, three 5's, three 6's, three 7's, and three 8 speed wheels respectively. Even if a 4th changes the ratio. The ratios are perfect example of composites. There are no primes, fractions, variables and betweens.

Bike stands are an ideal to see the symmetry. Viewing back wheels square on, you will see different dished like appearances in the back wheels. You will be looking at single, double and triple front chain wheel, with 1, 5, 6, or 7 rear sprocket sets. Mathematically pans out a statistic of 7 possible arrangements can be seen.

You will observe some bikes the gear cluster sides look dished and flat the opposite side. Others the gear cluster side look flat, while the opposite side looks dished. Still others both sides look flat while others, look dished. Mathematically speaking a statistic of 4 possible arrangements.

The distance across the rear wheel drop outs are perfect symmetry to acuminate the different length's of the hub speeds. The gear cluster and opposite sides of the hub flanges are corresponding flat or curved to center the rim mathematically right in the center of the stays.

What ever the dished/speed arrangement the hub perfectly centers wheels between the drop stays. You will observe plenty of even clearance both sides of the tires from the stays and under the brake bridges you can check with rulers. Observing dissembled hubs there will be corresponding curved or straight gear cluster and opposite side flanges.

Tires size is wheel size that mathematically determines spoke length. The size of the tire is written on the side of the tire wall. ( The x is the width of the tire ). A tape measure across a tire will confirm imperial inches and the flip side of the tape in metric.

Mountain and commuter bikes are common 26 inches, and metric equivalent 700mm, expressed as 700C. 26 inch and 700c wheels may look identical but comparing the two the metric wheels are a little smaller than the imperil equivalent.

Taking a measurement of the rim is rim 26 inch and 700C may look the same size first glance but the 700c is slightly smaller. If you have striped down rims you can see by comparing the 2.

Closely examining both front and rear wheel spoke arrangements on the hub flange, you will observe an alternating in and out spoke heads. Inward heads either the gear cluster or opposite side of the wheel or the outward heads and so on. Carefully note there is an alternating pair of spokes from the rim each side of the hub in a prefect repeating paten in the circumference of the wheel.

Observe spokes crossing heads of other spokes right on the flange. You will observe a curve either the gear cluster or opposite curved flange. The spoke crosses at an angle another spoke further up. Then another. This is called cross. Depending on your bike, racing bike wheels are commonly 2. Standard mountain and commuter bikes are 3 cross, and strong cargo carrying capability wheels is usually four.

The choice of hub is the most important part of building wheels.  The number of spoke holes, gives the number of spoke holes required by the rim. Either the gear cluster or opposite side flange is straight or curved will center the rim between the drop out stays.

Have a spare wheel on you to use as a model to assemble the spoke head paten.

Start with either side of the hub. Using the another wheel as a guide, insert the first spoke according to the in or outward head arrangement on either the gear cluster or the opposite side flange. It is best to start with the valve hole of the rim as a reference point. Insert the thread end of the spoke though the spoke hole screwing on the nipple.

Looking at the other wheel will tell you the next spoke to miss a whole in the hub flange inserting the spoke so the head in or out according to the first spoke. The model wheel will tell you to count 3 rim spoke holes along from the first spoke threading the spoke into the 5th hole screwing on the nipple. Check there is 3 spoke holes between the 2 spokes the second spoke 5 spoke holes along.

Repeat this completing with the next 5 spokes along completing the same side. Make sure all the spoke heads in the flange are all uniform in every empty spoke whole between the spokes are every 5th spoke hole along that there is 3 spoke holes between every spoke.

It is crucial not to miss the mathematical paten. It will cause an unbalanced string when completed. When you come to tension the spokes the unbalanced string will buckle the wheel. Tightening will pull a bow in the rim and cause a bump in the opposite corner creating a distortion in the rim.

No matter what you do you will make things worse creating a horrible horrible whole. The spoke heads must be all inwards with an empty spoke hole every between a spoke every 5th spoke hole along 3 empty spoke holes between every spoke.

Next thread a spoke in a empty spoke hole, so the head is opposite.( the other heads are in these are outwards ) of the spoke heads just threaded. Count, including the threaded spokes a total of 11 along threading the spoke thread into the 11th spoke hole. Make sure the spoke crosses the spoke on the flange.

It is crucial not to miss this paten. The spoke won't cross the spoke properly causing another unbalanced string when the wheel is completed. Again when you come to tension the spokes the unbalanced string will buckle the wheel. Like the other, when it comes to tightening spokes will pull a bow in the rim and cause a bump in the opposite corner creating a distortion in the rim you will never be able to straighten out.

Here you should have alternating 2 empty spoke holes for the other side left. Checking the symmetry of the other wheel will tell you to insert the next spoke according to the in and out head paten threading the spoke into the 5th rim hole along. The symmetry of the other wheel will tell you it repeats round the circumference of the wheel you assembling.

Once completed here you should have one last remaining spoke hole to finish off that side. If you followed the symmetry spoke paten of the other wheel to the letter the last remaining hole should be the automatically be the 11th hole from the spokes you just assembled.

When you come to adjust the spokes don't wrench the new nipple with the spoke tool. Only finger tight. It is important to remember you will be pulling a bow this part of the rim. You will be at risk of running out of thread to tight far to loose, or to loose with to much thread left over.

Rotate the wheel freely observing a side any side wobble between the brake pads. Don't wrench tight the new spoke any more than have too. You will great a one side wobble that needs undoing Reset back to were it was.

Don't wrench the new spoke several turns or you will start the process of creating get rid side to side whole. Just to a point they are firm.

Provided you have a well uniformed spoke cross round the whole wheel you should have a rim that runs straight and true though the brake rubbers.

All that needs to be done now is to remove the wheel and reassemble the tire. Don't inflate because the brake pads catching on the expanded tire wall will prevent you from fitting the wheel. The tire can be pumped up in the bike. Or alternatively release the brake cable to expand the brake arms clearing the fully inflated tire. The disadvantage of this you have to rest the brake setting.