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In Search of the Quad Axel

by George S. Rossano

Some definitions of terms we use for jumps in this article

Idealized Jump: A jump that is fully rotated in the air, with no missing rotation at the takeoff and landing, executed with the correct takeoff and landing edges.

True Jump:  For triple Lutz and below - the same as an idealized jump, as these are achievable as idealized jumps within the demonstrated capabilities of competitive skaters.  For triple Axel and above - a jump with no more than 1/4 rotation missing on the takeoff and no rotation missing at the landing, with correct takeoff and landing edges. (These jumps require some pre-rotation to achieve the necessary initial angular momentum to complete the jump, or so we currently believe).

Pseudo Single, Double, Triple, etc:  A jump missing more rotation than a true jump. (Pseudo here refers to the actual number of rotations not whether it is a jump.)

These definitions are for the discussion of the physics of jumps and are unrelated to the rules for the calling and scoring of jumps.

(6 January 2020)  From time to time over the past few seasons, Yuzuru Hanyu has spoken of the goal to acquire the quad Axel, most recently at the Grand Prix Final.  Several skaters have now reach quad Lutz.  Is the quad Axel next?

Is it even possible?

Several elite coaches I have asked that question this season and last all responded without the least hesitation, an emphatic yes.

But come on,  quad Axel?  Really?

The history of sport is filled with claims that some athletic feat was unobtainable, only in the end to be proven wrong.  That small detail, however, will not prevent us from taking a theoretical look at jumps to see if the quad Axel is achievable.

For idealized jumps, toe loop through Lutz jumps have an integer number of rotations in the air, while Axels have an integer number plus one-half.  For an attempt at a jump to be successful the skater must develop enough time in the air and enough angular momentum at the takeoff to complete the rotation.

The time is the air is determined by the height of the jump.  The jumping performance of elite skaters, and studies of counter movement jumps of elite athletes in training for various sports show that the maximum height the vast majority of elite athletes can achieve is about 29-30 inches.  Uncommonly, slightly greater heights can be achieved, though in the study of many jumps over the years, we have yet to verify a jump in competition over 30 inches; nevertheless, we accept the possibility.  Thus, for this study, we will assume that the maximum time an elite skater can achieve in the air is 0.8 seconds, corresponding to a height of 30.89 inches.  But based on what we have actually measured, the occasional jump a bit over 0.75 sec, heights top out at 29-30 inches for elite skaters.

Measuring angular momentum from imaging jumps is more work than we care to take on, but is not necessary for this study.  What we really care about is the angular speed of the rotation corresponding to the angular momentum.  Again, studying many jumps, we find elite skaters typically take off with angular speeds ranging from 2 to 3 rotations per second.  By adjusting their moment of inertia in-flight they achieve peak rotations rates of up to 6 rotations a second.  We have seen a few jumps where a peak speed of greater than 6 rotations per second may have been reached, perhaps as high as 6.25 rotations per second, but we can't say for certain.  For this study we will assume 6 rotations a second is the peak.  Skaters also typically land their jumps in the range of 2 to 3 rotations per second, while some elite men doing quads are capable of holding their landing at 4 rotations per second.

For idealized jumps we can generate the following table that gives the average rotation speed needed to achieve the required number of rotations in the air for idealized jumps.

      Single Double Triple Quad Quint
Time Height 1.00(1) 1.50(1) 2.00 2.50 3.00 3.50 4.00 4.50 5.00
(sec) (in)   T-Lz A T-Lz A T-Lz A T-Lz A T-Lz
0.05 0.12 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00
0.10 0.48 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00
0.15 1.09 6.67 10.00 13.33 16.67 20.00 23.33 26.67 30.00 33.33
0.20 1.93 5.00 7.50 10.00 12.50 15.00 17.50 20.00 22.50 25.00
0.25 3.02 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 20.00
0.30 4.34 3.33 5.00 6.67 8.33 10.00 11.67 13.33 15.00 16.67
0.45 9.77 2.22 3.33 4.44 5.56 6.67 7.78 8.89 10.00 11.11
0.50 12.07 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00
0.55 14.60 1.82 2.73 3.64 4.55 5.45 6.36 7.27 8.18 9.09
0.60 17.37 1.67 2.50 3.33 4.17 5.00 5.83 6.67 7.50 8.33
0.65 20.39 1.54 2.31 3.08 3.85 4.62 5.38 6.15 6.92 7.69
0.70 23.65 1.43 2.14 2.86 3.57 4.29 5.00 5.71 6.43 7.14
0.75 27.15 1.33 2.00 2.67 3.33 4.00 4.67 5.33 6.00 6.67
0.80 30.89 1.25 1.88 2.50 3.13 3.75 4.38 5.00 5.63 6.25

Table 1:  Average Rotation Speed Required for Idealized Jumps with Air Time up to 0.8 sec

The shaded cells are idealized jumps that cannot be achieved with a maximum rotation rate of 6 rotations per second.  We exclude the cells at 6.00, as the average rate must be less than the peak rate since the peak rate (assumed to be 6 rotations per second)  is never sustained for the entire air time of a jump.

The final piece of the puzzle is the relationship between the average rotation rate in the air and the maximum  rotation rate in the air which we assume to be 6 rotations per second.

Skaters typically take off with 2-3 rotations per second of angular speed, and adjust their moment of inertia in-flight to increase the rotations speed.  Some part of the time in the air is spent approaching their best air position, followed by a longer interval where small changes in rotation speed still take place as they make minor adjustments, and finally during checkout, their position opens up and their angular rotation speed decreases over a shorter time interval prior to the landing.  The goal of the skater is to get into the tightest position possible as soon as they can, hold near that position for as long as they can, and then reduce their rotational speed to a manageable rate for the landing, typically 2 to 3 rotations per second or less.  Approaching three rotations per second and above at the landing, many skaters cannot control the landing, and will step out of the landing, turn out, or fall. (2)

Of the three phases in-flight (pulling in, mid-flight and opening up), the time in each phase shows a lot of variety from one skater to the next and one jump type to the next.  Skaters get into a tight air position most quickly for Lutz and flip, less so for loop, even less for Salchow and toe loop, and most slowly for Axel.(3)

Based on studying a large number of jumps we find a reasonable assumption for average rotation speed is that for toe loop through Lutz the best skaters can do is achieve an average rotation rate that is 90% of the peak rotation rate, and for Axel the best they do is 80% of the peak rotation rate.  For most jumps however, skaters don't perform nearly this well.  The ratio of the average rotation rate to the peak rotation rate we will refer to as the air rotation efficiency.

Using the above values we can convert Table 1 from a table of average rotation rate to maximum rotation rate needed, for jumps of different height, assuming air rotational efficiency of 90% for toe loop through Lutz, and 80% for Axel.

      Single Double Triple Quad Quint
Time Height 1.00(1) 1.50(1) 2.00 2.50 3.00 3.50 4.00 4.50 5.00
(sec) (in)   T-Lz A T-Lz A T-Lz A T-Lz A T-Lz
0.05 0.12 22.22 37.50 44.44 62.50 66.67 87.50 88.89 112.50 111.11
0.10 0.48 11.11 18.75 22.22 31.25 33.33 43.75 44.44 56.25 55.56
0.15 1.09 7.41 12.50 14.81 20.83 22.22 29.17 29.63 37.50 37.04
0.20 1.93 5.56 9.38 11.11 15.63 16.67 21.88 22.22 28.13 27.78
0.25 3.02 4.44 7.50 8.89 12.50 13.33 17.50 17.78 22.50 22.22
0.30 4.34 3.70 6.25 7.41 10.42 11.11 14.58 14.81 18.75 18.52
0.45 9.77 2.47 4.17 4.94 6.94 7.41 9.72 9.88 12.50 12.35
0.50 12.07 2.22 3.75 4.44 6.25 6.67 8.75 8.89 11.25 11.11
0.55 14.60 2.02 3.41 4.04 5.68 6.06 7.95 8.08 10.23 10.10
0.60 17.37 1.85 3.13 3.70 5.21 5.56 7.29 7.41 9.38 9.26
0.65 20.39 1.71 2.88 3.42 4.81 5.13 6.73 6.84 8.65 8.55
0.70 23.65 1.59 2.68 3.17 4.46 4.76 6.25 6.35 8.04 7.94
0.75 27.15 1.48 2.50 2.96 4.17 4.44 5.83 5.93 7.50 7.41
0.80 30.89 1.39 2.34 2.78 3.91 4.17 5.47 5.56 7.03 6.94

Table 2:  Peak Rotation Speed Required for Idealized Jumps with Air Time up to 0.8 sec

Table 2 assumptions:

  • Maximum peak rotation rate of 6 rotations per second
  • Maximum in-flight rotation efficiency of 90% for toe loop through Lutz
  • Maximum in-flight rotation efficiency of 80% for Axel
  • Maximum time in the air 0.80 seconds

For our stated assumptions, an idealized quad Axel is not achievable.  Table 2 also says that quads should be less common than is currently the case.  What makes them more common than this table might suggest is that quads all tend to have missing rotation, making them easier to complete.

Keep in mind, the values in Table 2 are not the only way to achieve a given jump, as there is a trade built into the table.  Consider for example a double Axel with 0.6 seconds in the air.  Table 2 says to achieve that jump a peak rotation rate of 5.21 rotations per second would be needed with an air rotation efficiency of 0.8.  The skater, however, would also achieve that jump with a higher peak rotation rate and a lower air rotation efficiency.  At a peak rotation rate of 6 rotations per second the air rotation efficiency could be as low as 0.695 and the jump would still be fully rotated.  So within certain limits, a less efficient air position can be compensated for by increased peak rotation rate.

So how inefficient can a jump be and still be fully rotated?  We recast Table 2 to show the lowest air rotation efficiency for a peak rotation rate of 6 rotations per second, that will fully rotate a jump.

      Single Double Triple Quad Quint
Time Height 1.00(1) 1.50(1) 2.00 2.50 3.00 3.50 4.00 4.50 5.00
(sec) (in)   T-Lz A T-Lz A T-Lz A T-Lz A T-Lz
0.05 0.12 3.33 5.00 6.67 8.33 10.00 11.67 13.33 15.00 16.67
0.10 0.48 1.67 2.50 3.33 4.17 5.00 5.83 6.67 7.50 8.33
0.15 1.09 1.11 1.67 2.22 2.78 3.33 3.89 4.44 5.00 5.56
0.20 1.93 0.83 1.25 1.67 2.08 2.50 2.92 3.33 3.75 4.17
0.25 3.02 0.67 1.00 1.33 1.67 2.00 2.33 2.67 3.00 3.33
0.30 4.34 0.56 0.83 1.11 1.39 1.67 1.94 2.22 2.50 2.78
0.45 9.77 0.37 0.56 0.74 0.93 1.11 1.30 1.48 1.67 1.85
0.50 12.07 0.33 0.50 0.67 0.83 1.00 1.17 1.33 1.50 1.67
0.55 14.60 0.30 0.45 0.61 0.76 0.91 1.06 1.21 1.36 1.52
0.60 17.37 0.28 0.42 0.56 0.69 0.83 0.97 1.11 1.25 1.39
0.65 20.39 0.26 0.38 0.51 0.64 0.77 0.90 1.03 1.15 1.28
0.70 23.65 0.24 0.36 0.48 0.60 0.71 0.83 0.95 1.07 1.19
0.75 27.15 0.22 0.33 0.44 0.56 0.67 0.78 0.89 1.00 1.11
0.80 30.89 0.21 0.31 0.42 0.52 0.63 0.73 0.83 0.94 1.04

Table 3:  Minimum Air Rotation Efficiency Required for Idealized Jumps with a Peak Rotation Rate of 6 Rotations per Second.

Keep in mind, Table 3 is for a peak rotation rate of 6 rotations per second.  For higher peak rotation rates, should they become obtainable, the minimum air rotation efficiency would be lower.

Note also that for an idealized quad Axel a rotation efficiency of 0.94 would be needed for a peak rotation rate of 6 rotations per second, an efficiency skaters are unlikely to ever achieve for an Axel type jump.

Jumps are not Really Ideal

The calculations to this point are for idealized jumps.  We call them idealized jumps because in the real world most skaters execute jumps with technique different from the ideal descriptions, and for triple Axel and above very different.  Further, certain jumps even when executed in textbook fashion never have had an integer number of rotations.  Loops and Salchow, for example, have always been executed with about one-quarter rotation short of integer rotation, as the skaters jump not only upwards, but also somewhat to the side for these jumps

When jumps are not fully rotated the ISU terminology is that the jump has "missing rotation."  Colloquially these are referred to as "cheated" jumps.  Jumps are designated  "under-rotated" if they are missing 1/4 rotation or more, and "downgraded" if they are missing 1/2 rotation or more.  The rules do not specify from where the rotation is missing.  A jump can be missing rotation from either the takeoff or the landing, or both (doubly cheated).

When a jump is missing rotation at the takeoff, the ISU does not currently under or downgrade the jump, no matter how egregious the cheat.  A forward takeoff on a toe loop (called a toe Axel) was occasionally called under or down, but this no longer seems to be the case, again no matter how egregious the cheat.  In the U.S. toe Axels are still regularly called, but toe Axel is the only jump with missing rotation on the takeoff that ever seems to be called in the U.S.

Looking at ISU competition specifically, many skaters have up to one-half rotation missing on the takeoff and up to one quarter rotation on the landing, and ISU Technical Panels will call those jumps fully rotated, giving full base value.  This means a full value jump can often be missing a total of three-quarters of a rotation for toe loop through Lutz, and one-half for Axels (Axels tend to have no more than one-quarter missing rotation on the takeoff, one-third at worst).  This is the method the Russian wunderkind (and others) now use to get credit for triple Axel and quads.  Most Russian girls do it the same way, and as the most successful ones are all taught by the same coach, it appears this is the technique they are taught, and not a coincidence, knowing the ISU will not penalize them.

With the above in mind, we recalculate Table 2 for two cases, jumps that have missing rotation on the takeoff only (Table 4), and jumps that have missing rotation on both the takeoff and landing (Table 5).   Our assumptions for the maximum capabilities of elite athletes remain the same.

      Single Double Triple Quad Quint
Time Height 0.50(1) 1.25(1) 1.50 2.25 2.50 3.25 3.50 4.25 4.50
(sec) (in)   T-Lz A T-Lz A T-Lz A T-Lz A T-Lz
0.05 0.12 11.11 31.25 33.33 56.25 55.56 81.25 77.78 106.25 100.00
0.10 0.48 5.56 15.63 16.67 28.13 27.78 40.63 38.89 53.13 50.00
0.15 1.09 3.70 10.42 11.11 18.75 18.52 27.08 25.93 35.42 33.33
0.20 1.93 2.78 7.81 8.33 14.06 13.89 20.31 19.44 26.56 25.00
0.25 3.02 2.22 6.25 6.67 11.25 11.11 16.25 15.56 21.25 20.00
0.30 4.34 1.85 5.21 5.56 9.38 9.26 13.54 12.96 17.71 16.67
0.45 9.77 1.23 3.47 3.70 6.25 6.17 9.03 8.64 11.81 11.11
0.50 12.07 1.11 3.13 3.33 5.63 5.56 8.13 7.78 10.63 10.00
0.55 14.60 1.01 2.84 3.03 5.11 5.05 7.39 7.07 9.66 9.09
0.60 17.37 0.93 2.60 2.78 4.69 4.63 6.77 6.48 8.85 8.33
0.65 20.39 0.85 2.40 2.56 4.33 4.27 6.25 5.98 8.17 7.69
0.70 23.65 0.79 2.23 2.38 4.02 3.97 5.80 5.56 7.59 7.14
0.75 27.15 0.74 2.08 2.22 3.75 3.70 5.42 5.19 7.08 6.67
0.80 30.89 0.69 1.95 2.08 3.52 3.47 5.08 4.86 6.64 6.25

Table 4:  Peak Rotation Speed Required for Jumps with Missing Rotation on Takeoff Only

      Single Double Triple Quad Quint
Time Height 0.25(1) 1.00(1) 1.25 2.00 2.25 3.00 3.25 4.00 4.25
(sec) (in)   T-Lz A T-Lz A T-Lz A T-Lz A T-Lz
0.05 0.12 5.56 25.00 27.78 50.00 50.00 75.00 72.22 100.00 94.44
0.10 0.48 2.78 12.50 13.89 25.00 25.00 37.50 36.11 50.00 47.22
0.15 1.09 1.85 8.33 9.26 16.67 16.67 25.00 24.07 33.33 31.48
0.20 1.93 1.39 6.25 6.94 12.50 12.50 18.75 18.06 25.00 23.61
0.25 3.02 1.11 5.00 5.56 10.00 10.00 15.00 14.44 20.00 18.89
0.30 4.34 0.93 4.17 4.63 8.33 8.33 12.50 12.04 16.67 15.74
0.45 9.77 0.62 2.78 3.09 5.56 5.56 8.33 8.02 11.11 10.49
0.50 12.07 0.56 2.50 2.78 5.00 5.00 7.50 7.22 10.00 9.44
0.55 14.60 0.51 2.27 2.53 4.55 4.55 6.82 6.57 9.09 8.59
0.60 17.37 0.46 2.08 2.31 4.17 4.17 6.25 6.02 8.33 7.87
0.65 20.39 0.43 1.92 2.14 3.85 3.85 5.77 5.56 7.69 7.26
0.70 23.65 0.40 1.79 1.98 3.57 3.57 5.36 5.16 7.14 6.75
0.75 27.15 0.37 1.67 1.85 3.33 3.33 5.00 4.81 6.67 6.30
0.80 30.89 0.35 1.56 1.74 3.13 3.13 4.69 4.51 6.25 5.90

Table 5:  Peak Rotation Speed Required for Jumps with Missing Rotation on Takeoff and Landing

From Tables 4 and 5 we note the following:

  • With lesser rotation in the jumps, the missing-rotation jumps can be achieved with lower rotation rates, and hence less air time and less height.
  • Pseudo-quads can be achieved with less height, and do not require near maximum height for true quads.  This increases the prevalence of quads in competition.
  • A Quint is more likely to be achieved than a quad Axel since the air rotation efficiency for toe loop through Lutz is greater than for an Axel.
  • Quad Axel would require the ability to exceed the assumptions on which these tables are based; for example, the ability to reach a maximum rotation rate of 6.25 per second or greater, or an air rotation efficiency significantly greater than 80%, or an air time significantly greater than 0.8 sec.

Quad Axel?  Probably Not

Given the skill level needed to achieve a fully rotated quad Axel (Table 2), it seems unlikely we will ever see one of these. It would probably require a revolutionary improvement in technique, training and/or equipment for a skater to outperform the assumptions built into Table 2, particular the time it takes to get over the right side and pulled in; i.e., an air rotation efficiency approaching 0.9.  A dynamical model of a true quad Axel can be found here.  A dynamical model of a true quad Lutz can be found here.

Will there be a skater that pushes the limits on the assumptions in Table 5?  History says we should be hesitant to say never, but even if a pseudo-quad Axel is ever landed, is it really a quad Axel?

We end with a summary of whether revolutionary improvements in technique can advance skating to true quad Axels and quints.  To do that, skaters and coaches will have to find ways to shatter limitations in one or more of the following ways.

Increase jump heights beyond current capability, to increase air time.  One obstacle to this is that to increase air time by a factor X, jumps heights must increase by X2.  You want to increase air time by twenty percent? You have to increase jump height by over a foot.

Push air rotation efficiency to extreme limits.  The challenge here is that the pulling in and opening up phases are already extremely short.  There isn't much there to shave off these times.

Increase the angular momentum at takeoff.  This could be by increasing the takeoff rotation rate with current moments of inertia, or taking off at current rotation rates with more open positions.  (This would allow a greater reduction of the moment of inertia in the air with a corresponding increase in peak rotation rate.)  The challenger here is how to do that in the first place, and then without introducing missing rotation, and in a way that keeps the jump under control.

Where is Casey Carlyle when we need her?(4)

Notes:

(1) For single double triple, etc. the first column is air rotations for toe loop through Lutz, and the second is air rotations for Axel.  Quint Axel is not included in the tables.

(2) Skaters with poor air positions, and thus reduced rotation rate often try to make up for this by holding the mid-flight position as long as possible, and to make the opening-up phase as short as possible.  The frequent result is a jump landed at too high a rotation rate to control.

(3) Skaters who struggle with triple Axel can take as much as one half their air time to get close to a tight air position, resulting in an air rotation efficiency too low to make completing the jump possible.

(4) So you don't have to look it up, that's an "Ice Princess" reference.

Copyright 2020 by George S. Rossano